WASHINGTON — Former Vice President Mike Pence turned up in Hungary last month to speak to a conference on conservative social values hosted by the far-right government of Prime Minister Viktor Orban.

Jeff Sessions, the former attorney general, was another recent visitor. Tucker Carlson did his Fox News show from Hungary for a week this summer. The American Conservative Union is planning a version of its CPAC gathering in Budapest early next year.

Those are among the more visible recent fruits of a well-funded campaign by Mr. Orban in the United States that stretches back a decade and now stands as a case study in how governments around the world seek to shape policies and debates in Washington, sometimes raising concerns about improper foreign influence in U.S. politics.

Carried out by a network of government offices, Washington lobbyists, Hungarian diaspora groups, educational institutions and government-funded foundations, the effort’s main impact has been to bolster Mr. Orban’s image as a conservative leader on the world stage — and to counter his reputation as an authoritarian nationalist who is cozying up to Russia and China.

It has also notched some tangible, if fleeting, policy victories for Mr. Orban, including the withdrawal during the Trump administration of a State Department grant to nurture independent news media in Hungary and the securing of a long-coveted Oval Office meeting for Mr. Orban in 2019 with President Donald J. Trump.

Much of what the Hungarian network has done is legal and standard operating procedure in Washington. But some of its activities touch on gray areas, including transparency requirements for those acting on behalf of foreign interests, concerns arising from overseas efforts to sway presidential campaigns, and the ethics of think tanks accepting money from governments or their proxies.

Last year, two prominent foreign policy think tanks in Washington severed ties with the Hungary Foundation, a group funded by the Hungarian government, amid concerns about its connections.

And Joseph R. Biden Jr.’s presidential campaign distanced itself from a volunteer policy adviser after receiving reports of his ties to Mr. Orban. The campaign referred the matter to the F.B.I., according to people familiar with the situation.

The network’s efforts can be traced to around 2011, the year after Mr. Orban became prime minister for the second time. As he worked to raise his global profile, he convened people with ties to Hungary from around the world in a Diaspora Council partly “to assist in developing a coordinated representation of the Hungarian position and interests in foreign media and toward decision makers,” according to government documents.

Among the groups represented on the roster of attendees at the council’s first meetings were U.S. organizations that have played roles in the influence network.

The leader of one group that participated in the council, Maximilian N. Teleki, has been active for years in Democratic politics. He spearheaded an informal Hungarian-Americans for Biden effort, donated $1,715 to the campaign and served on its volunteer foreign policy group on Europe.

In August 2020, though, Brian P. McKeon, who was serving as a foreign policy adviser to the Biden campaign and is now a top official in the State Department, informed Mr. Teleki that the campaign had “received several reports and expressions of concern about your having connections” to Mr. Orban’s government and political party, Fidesz, according to emails obtained by The New York Times.

Mr. McKeon wrote that the campaign was “not in a position to determine the veracity of these reports” but added “we think it best, for the good of the campaign, that you step back from your participation” in the foreign policy advisory network, which included hundreds of volunteers.

The warning came as Mr. Biden’s campaign was taking precautions to avoid comparisons to 2016, when Russia interfered to try to help Mr. Trump in the election.

“I can unequivocally state that I have had no substantive contacts with ANY members of the Hungarian government or Fidesz since I started working” on the advisory group, Mr. Teleki replied, distancing himself from Mr. Orban.

Mr. Teleki stepped away from the policy group, and campaign officials reported their concerns to the F.B.I., according to people familiar with the situation. It is not clear whether there is an active investigation.

The episode did not stop Mr. Teleki from occasionally engaging with the Biden team, including helping arrange a campaign policy briefing on Zoom for people with ties to Central and Eastern European countries.

Last month, Mr. Teleki joined an off-the-record call hosted by the White House, during which he asked a question about the treatment of ethnic minorities in Ukraine — an issue that Mr. Orban has used as a cudgel against Ukraine.

Another prominent figure in the Hungarian diaspora, Frank Koszorus Jr., said there had been discussions about efforts to try to improve the bilateral relationship under Mr. Biden.

“People are interested in contacts with Democrats as well as Republicans,” said Mr. Koszorus, who had registered to lobby for the Hungarian embassy in the 1990s and the early 2000s.

He also served until recently as the chairman of the American Hungarian Federation. The group pressured Mr. Trump’s administration to reconsider the State Department grant intended to foster free press in Hungary. Mr. Orban’s allies had criticized the grant as an example of Washington meddling in their affairs.

To finance some of the efforts in the United States, Mr. Orban’s government authorized the creation and funding of a nonprofit group in 2012 that would come to be known as the Hungary Foundation.

It has donated more than $5.2 million through the end of last year to think tanks, conservative groups, colleges and Hungarian-American organizations — including $1.3 million to groups previously led by Mr. Teleki and Mr. Koszorus — to fund conferences, fellowships and cultural programs.

In 2014 and 2015, some of the foundation’s more prominent board members, including former Gov. George E. Pataki of New York and the diplomat Kurt D. Volker, stepped down over the possibility that the group might be required to register as an agent of the Hungarian government under the Foreign Agents Registration Act, or FARA. The act requires the disclosure of lobbying or public relations campaigns in the United States on behalf of certain foreign interests.

In late 2015, the Justice Department issued an advisory opinion that the foundation could be exempt from registering under FARA as long as its activities were limited to “bona fide religious, scholastic, academic or scientific pursuits or of the fine arts,” and it did not engage in lobbying or public relations.

Mr. Volker and Mr. Pataki rejoined the board in 2016, and the foundation hired a new executive director, Anna Smith Lacey, a former congressional liaison in the Hungarian Embassy in Washington.

The foundation and Ms. Smith Lacey were soon engaging with U.S. officials and policy debates in ways that sometimes seemed to test the language in the Justice Department’s opinion.

Ms. Smith Lacey, a Hungarian citizen whose father is a prominent government official there, attended one of a series of meetings about Central Europe at the State Department that a Trump appointee held with representatives of groups focused on the region. And she appeared at exclusive gatherings with U.S. officials overseeing Central Europe organized by recipients of foundation grants, including the Atlantic Council and the Center for European Policy Analysis, each of which had received more than $200,000 from the Hungary Foundation.

At a panel hosted by the Atlantic Council in 2019, Ms. Smith Lacey pushed back against claims that democratic governance was eroding in Hungary and accused the United States and other Western countries of hypocrisy, asserting that they treated immigrants, Jews and Christians poorly. She linked Mr. Orban’s agenda to the American nationalism preached by Mr. Trump.

Ms. Smith Lacey said her attendance at the State Department meeting was unrelated to her role with the foundation, and that the foundation grants did not support the Atlantic Council conference.

But in 2019, the Atlantic Council and the Hungary Foundation entered a partnership to hold private strategy retreats with U.S. and Hungarian policymakers and experts.

A delegation from the Atlantic Council traveled to Budapest a few months after the conference to lay the groundwork for the retreats, which the council envisioned as a platform to discuss issues including the erosion of the rule of law and press freedom.

But a meeting at the foreign ministry grew confrontational, with Hungarian officials insisting that the topics be limited to less contentious issues like trade and energy, according to the Atlantic Council. The Hungarian government did not respond to a request for a comment.

After the meeting, Ms. Smith Lacey’s foundation notified the council that the Hungarian government was no longer interested in the project.

“Our board has decided that the project was not a good fit for us due to FARA considerations,” Ms. Smith Lacey said in an email to The Times.

But Daniel Fried — a former U.S. ambassador to Poland who was on the trip as an Atlantic Council scholar — said he was left with the impression that the change of heart occurred high up in Mr. Orban’s government.

“The question is, and it’s a fair question, did they want to move to the center and have us help sort of validate that kind of move, or did they want to stay on the hard right and simply use us as cover?” Mr. Fried said in an interview. “If Orban was interested in finding a way back to a center-right place, this would have been one way he could do it. But in my view, he wasn’t interested. He was going to go all in on hard right.”

Last year, the Atlantic Council returned a $158,000 grant for the retreats and ended its relationship with the Hungary Foundation, which has been funded almost entirely by $21 million in grants from the Hungarian government, according to Hungarian and U.S. government documents.

The Center for European Policy Analysis also ended its relationship with the foundation amid concerns about its ties to Mr. Orban’s government as well as a potential conflict between Mr. Volker’s roles as a board member of the foundation and a fellow at the center, according to people familiar with the situation.

Ms. Smith Lacey rejected the idea that the foundation was part of a Hungarian government influence operation, saying “our mission is cultural, educational and scholastic. We are not a lobby. Hungary has lobbying firms. I am not interested in that at all.”

Since May 2012, Mr. Orban’s government has paid more than $4.5 million to lobbyists.

His office paid $83,000 in 2013 and 2014 to a well-connected Washington lobbying firm that had partnered with a consulting firm managed by a former cabinet minister in his government who was serving as the chief executive of the Hungary Foundation.

In late 2014, Mr. Orban’s office hired Connie Mack IV, a Republican from Florida who had served in Congress until 2013.

Over the next few years, Mr. Orban’s government paid $3.4 million to firms run by Mr. Mack, including one that partnered with an Orban-backed Hungarian think tank.

Mr. Mack helped win support for withdrawing the State Department media grant that Mr. Orban’s allies had opposed.

After Mr. Mack’s contract ended, the Hungarian government signed a deal with Policy Impact Strategic Communications, a firm run by William Nixon, a veteran Republican lobbyist, that was paid $265,000 to provide lobbying and public relations assistance in 2018 and 2019. It had previously helped form a group called the Hungarian American Institute that arranged a visit by Hungarian legislators to Washington.

The father of Mr. Carlson, the Fox News host, is on the board of both Policy Impact and the Hungarian American Institute. The elder Mr. Carlson does not appear to have been involved in his son’s coverage of Mr. Orban’s government.

But Policy Impact helped lay the groundwork. The firm introduced the younger Mr. Carlson to the Hungarian ambassador, arranged an interview on Mr. Carlson’s show in 2019 with Hungary’s minister of foreign affairs and trade, and helped to arrange meetings with Hungarian officials for a Fox News producer who traveled to Budapest in 2019 to research Mr. Orban’s agenda.

Asked about the lobbying firm’s efforts, Mr. Carlson said in a text message that “none of this is relevant to anything,” adding that he interviewed Mr. Orban because he was impressed by the Hungarian leader’s immigration policies.

Another organization that has received funding from the Hungarian government, the Danube Institute, has provided fellowships for Western conservatives and also arranged and paid for a speech in Budapest last month by Mr. Sessions, who said he was promised a modest fee.

Mr. Pence lauded Mr. Orban’s social agenda during his speech, but he also urged a tough stance against China, an uncomfortable issue given Mr. Orban’s increasing alignment with Beijing. His appearance came at a conference hosted by Hungary’s minister for families.

The minister was among the Hungarians who met in Washington with officials from the American Conservative Union as the group considered an installment of its Conservative Political Action Conference in partnership with the Budapest-based Center for Fundamental Rights, another organization linked to the Hungarian government.

“It certainly seems like the government wants to partner with us,” said Matt Schlapp, the president of the conservative union.