As a federal investigation into Rudolph W. Giuliani escalates, his advisers have been pressing aides to former President Donald J. Trump to reach into a $250 million war chest to pay Mr. Giuliani for his efforts to overturn the results of the 2020 election on Mr. Trump’s behalf.

The pressure from Mr. Giuliani’s camp has intensified since F.B.I. agents executed search warrants at Mr. Giuliani’s home and office last week, according to people familiar with the discussions, and comes as Mr. Giuliani has hired new lawyers and is facing his own protracted — and costly — legal battles.

Federal prosecutors in Manhattan have been examining communications between Mr. Giuliani, Mr. Trump’s former personal lawyer, and Ukrainian officials as he tried to unearth damaging information about President Biden before the election. The prosecutors are investigating whether Mr. Giuliani lobbied the Trump administration on behalf of Ukrainian officials who were helping him, a potential violation of federal law.

Mr. Giuliani, who has not been charged, has denied any wrongdoing and denounced the searches as “corrupt.” The actions in Ukraine were part of Mr. Trump’s first impeachment trial.

Separately, Mr. Giuliani is being sued for defamation by two voting machine companies, Dominion and Smartmatic, for his false claims that the companies were involved in a conspiracy to flip votes to Mr. Biden.

Mr. Giuliani led the effort to subvert the results of the 2020 race in a series of battleground states, but he was not paid for the work, according to people close to both Mr. Giuliani and Mr. Trump. His supporters now want the Trump campaign to tap into the $250 million it raised in the weeks after the election to pay Mr. Giuliani and absorb costs he has incurred in the defamation suits.

“I want to know what the GOP did with the quarter of $1 billion that they collected for the election legal fight,” Bernard Kerik, the former New York City police commissioner, wrote on Twitter on Sunday. Mr. Giuliani appointed Mr. Kerik when he was mayor of New York.

Using expletives, Mr. Kerik added that “lawyers and law firms that didn’t do” much work were paid handsomely, while those who worked hard “got nothing.”

Mr. Kerik has made similar complaints to some of Mr. Trump’s advisers privately, according to people familiar with the conversations, arguing that Mr. Giuliani has incurred legal expenses in his efforts to help Mr. Trump and that Mr. Giuliani’s name was used to raise money during the election fight.

In a separate tweet, Mr. Kerik blamed the Republican National Committee chairwoman, Ronna McDaniel. R.N.C. officials said that the group did not make the same overt fund-raising appeals as the Trump campaign to challenge the election results.

A lawyer for Mr. Giuliani, Robert J. Costello, has had conversations with a lawyer for Mr. Trump about whether any of the material that was seized by the F.B.I. should be protected from scrutiny because of attorney-client privilege. Mr. Costello has also raised the question of paying Mr. Giuliani, according to two people briefed on those discussions.

Jason Miller, a spokesman for Mr. Trump, declined to comment. Mr. Giuliani could not be reached for comment.

Mr. Giuliani had encouraged Mr. Trump to file challenges to the election, and the former president tasked Mr. Giuliani with leading the effort in November. But when Mr. Giuliani’s associate, Maria Ryan, sent an email to Trump campaign officials seeking $20,000 a day for his work, Mr. Trump balked, The New York Times has reported.

Mr. Trump later told his advisers he did not want Mr. Giuliani to receive any payment, according to people close to the former president with direct knowledge of the discussions. Before Mr. Trump left the White House in January, he agreed to reimburse Mr. Giuliani for more than $200,000 in expenses but not to pay a fee.

Some of Mr. Giuliani’s supporters have blamed Mr. Trump’s aides — and not the former president — for the standoff. However, people close to Mr. Trump said he has stridently refused to pay Mr. Giuliani.

Mr. Giuliani’s advisers were also disappointed that he did not receive a federal pardon from Mr. Trump, despite facing the long-running federal investigation into his Ukrainian dealings, a person close to Mr. Giuliani said. After months of speculation that Mr. Trump might issue Mr. Giuliani a pre-emptive pardon, Mr. Giuliani said on his radio show in January that he did not need a pardon, because “I don’t commit crimes.”

The efforts to overturn the election culminated in a rally of Mr. Trump’s supporters near the White House on Jan. 6. After marching to the Capitol, where the Electoral College results were being certified, hundreds of those supporters stormed the building, resulting in deaths and scores of injuries to Capitol Police officers and others. The events led to Mr. Trump’s second impeachment trial, and Mr. Trump told Mr. Giuliani in a private meeting that he could not represent him in the proceedings, people briefed on the meeting said.

Asked about Mr. Kerik’s tweet during an interview with ABC News, Mr. Giuliani’s son, Andrew, said that his father’s fees should be covered by Trump’s campaign coffers.

“I do think he should be indemnified,” the younger Mr. Giuliani said. “I think all those Americans that donated after Nov. 3, they were donating for the legal defense fund. My father ran the legal team at that point. So I think it’s very easy to make a very strong case for the fact that he and all the lawyers that worked on there should be indemnified.”

He added, “I would find it highly irregular if the president’s lead counsel did not get indemnified.”

A person close to Mr. Giuliani, who was granted anonymity because this person was not authorized to discuss the matter publicly, made a related argument, saying the Trump campaign should be careful to ensure money in the war chest was spent in connection with the election effort because it was solicited from the public for that purpose.

Although there are many differences between the two situations, for some of Mr. Trump’s advisers, the standoff with Mr. Giuliani has raised uncomfortable echoes of a similar dispute with another of Mr. Trump’s former personal lawyers, Michael D. Cohen.

In 2019, Mr. Cohen said the Trump Organization, Mr. Trump’s family business, breached an agreement with him to cover his legal costs. In a lawsuit, Mr. Cohen said the company initially paid some of the bills after the F.B.I. searched his apartment and office in April 2018. But, he said in the lawsuit, company officials stopped the payments when they discovered around June 2018 that he was preparing to cooperate with federal investigators.

Mr. Cohen pleaded guilty later that year to charges related to tax evasion, as well as a campaign finance charge related to his 2016 hush-money payment to a pornographic film star who had claimed to have had an affair with Mr. Trump. Mr. Cohen ended up testifying about Mr. Trump in Congress, and provided assistance to the investigation led by the special counsel Robert S. Mueller III into possible conspiracy between the Trump campaign and Russian officials.

After the F.B.I. searched Mr. Cohen’s home and office, he filed a civil action against the U.S. attorney in Manhattan, which Mr. Trump joined to prevent federal officials from gaining access to material that could be protected by attorney-client privilege between Mr. Trump and Mr. Cohen.

Mr. Giuliani’s lawyers are considering filing a similar action in his case, according to one of the people close to the former mayor. One lawyer advising Mr. Giuliani, Alan Dershowitz, told CNN that it would be appropriate for Mr. Trump to join such an effort. Mr. Dershowitz confirmed the comment to The Times.

Mr. Giuliani recently added four new lawyers to his team: Arthur L. Aidala, a former Brooklyn prosecutor and former Fox News commentator; Barry Kamins, a retired New York Supreme Court justice and law professor; the retired New York Appellate Division Justice John Leventhal; and Michael T. Jaccarino, a former Brooklyn prosecutor.

William K. Rashbaum contributed reporting.