As the House select committee investigating the Jan. 6 assault on the Capitol rushes to gather evidence and conduct interviews, how far it will be able to go in holding former President Donald J. Trump accountable increasingly appears to hinge on one possible witness: former Vice President Mike Pence.

Since the committee was formed last summer, Mr. Pence’s lawyer and the panel have been talking informally about whether he would be willing to speak to investigators, people briefed on the discussions said. But as Mr. Pence began sorting through a complex calculation about his cooperation, he indicated to the committee that he was undecided, they said.

To some degree, the current situation reflects negotiating strategies by both sides, with the committee eager to suggest an air of inevitability about Mr. Pence answering its questions and the former vice president’s advisers looking for reasons to limit his political exposure from a move that would further complicate his ambitions to run for president in 2024.

But there also appears to be growing tension.

In recent weeks, Mr. Pence is said by people familiar with his thinking to have grown increasingly disillusioned with the idea of voluntary cooperation. He has told aides that the committee has taken a sharp partisan turn by openly considering the potential for criminal referrals to the Justice Department about Mr. Trump and others. Such referrals, in Mr. Pence’s view, appear designed to hurt Republican chances of winning control of Congress in November.

And Mr. Pence, they said, has grown annoyed that the committee is publicly signaling that it has secured a greater degree of cooperation from his top aides than it actually has, something he sees as part of a pattern of Democrats trying to turn his team against Mr. Trump.

For the committee, Mr. Pence’s testimony under oath would be an opportunity to establish in detail how Mr. Trump’s pressuring him to block the certification of Joseph R. Biden Jr.’s victory brought the country to the brink of a constitutional crisis and helped inspire the storming of the Capitol on Jan. 6, 2021.

It could also be vital to the committee in deciding whether it has sufficient evidence to make a criminal referral of Mr. Trump to the Justice Department, as a number of its members have said they could consider doing. The potential charge floated by some members of the committee is violation of the federal law that prohibits obstructing an official proceeding before Congress.

The combination of the pressure brought to bear on Mr. Pence and Mr. Trump’s repeated public exhortations about his vice president — “If Mike Pence does the right thing, we win the election,” he told supporters on the Ellipse just before they marched to the Capitol — could help the committee build a well-documented narrative linking Mr. Trump to the temporary halting of the vote certification through rioters focused, at his urging, on Mr. Pence.

A criminal referral from the committee would carry little legal weight, but could increase public pressure on the Justice Department. The department has given little indication of whether it is seriously considering building a case against Mr. Trump.

Attorney General Merrick B. Garland said last week that federal prosecutors remained “committed to holding all Jan. 6 perpetrators, at any level, accountable under law — whether they were present that day or were otherwise criminally responsible for the assault on our democracy.” But he did not mention Mr. Trump or indicate whether the department considered obstruction of Congress a charge that would fit the circumstances.

There are nonetheless some early indications that federal prosecutors working on charging the Capitol rioters are looking carefully at Mr. Trump’s pressure on Mr. Pence — and his efforts to rally his supporters to keep up that pressure even after Mr. Pence decided that he would not block certification of the Electoral College results.

In plea negotiations, federal prosecutors recently began asking defense lawyers for some of those charged in Jan. 6 cases whether their clients would admit in sworn statements that they stormed the Capitol believing that Mr. Trump wanted them to stop Mr. Pence from certifying the election. In theory, such statements could help connect the violence at the Capitol directly to Mr. Trump’s demands that Mr. Pence help him stave off his defeat.

Gina Bisignano, a Beverly Hills beautician who helped her fellow Trump supporters smash at a window at the Capitol, noted in court papers connected to her plea that she had marched on the building specifically after hearing Mr. Trump encourage Mr. Pence “to do the right thing.”

While in the crowd, the papers say, Ms. Bisignano filmed herself saying, “We are marching on the Capitol to put some pressure on Mike Pence.” The papers also note that once Ms. Bisignano reached the building, she started telling others “what Pence’s done,” and encouraged people carrying tools like hatchets to break the window.

Similarly, Matthew Greene, a member of the Central New York chapter of the Proud Boys, said in court papers connected to his own guilty plea that he had conspired with other members of the far-right group to “send a message to legislators and Vice President Pence” who were inside the Capitol certifying the final stage of the election.

“Greene hoped that his actions and those of his co-conspirators would cause legislators and the vice president to act differently during the course of the certification of the Electoral College vote than they would have otherwise,” the papers said.

Mr. Trump’s pressure campaign on Mr. Pence has been well established in news reports and books over the past year. Mr. Trump, aided at times by a little-known conservative lawyer, John Eastman, repeatedly pressured Mr. Pence to intervene in Congress’s certification of the 2020 presidential election, saying he had the power to delay or alter the outcome.

Mr. Pence consulted a variety of people in weighing what to do, and when he ultimately refused, Mr. Trump attacked him with harsh words.

Once the mob stormed the Capitol, with some rioters chanting for Mr. Pence to be hanged, Mr. Trump initially brushed aside calls from aides and allies to call them off.

In the last week, around the anniversary of the Jan. 6 riot, both the chairman of the committee, Representative Bennie Thompson, Democrat of Mississippi, and its vice chairwoman, Representative Liz Cheney, Republican of Wyoming, have suggested they want Mr. Pence to testify voluntarily.

On Friday, Mr. Thompson told NPR that the committee might issue Mr. Pence a formal invitation as soon as the end of the month. That same day, another committee member, Representative Adam B. Schiff, Democrat of California, underlined Mr. Pence’s importance in a television interview, saying he viewed him “as an indispensable person to talk to.”

A refusal by Mr. Pence to cooperate could lead the committee to take the highly unusual move of subpoenaing a former vice president, setting up a potential court fight that could delay a resolution for months as the committee tries to wrap up its work before the election.

Mr. Pence’s personal lawyer, Richard Cullen, began discussions this summer with the top investigator on the House Jan. 6 committee, Timothy Heaphy, a former federal prosecutor. Mr. Cullen had worked alongside Mr. Heaphy at the same law firm several years ago.

Since then, the committee has declined to formally ask Mr. Pence for an interview and Mr. Cullen has told the committee he is unsure what Mr. Pence will do. Both sides hoped that, given Mr. Cullen’s relationship with Mr. Heaphy, they could work out some sort of agreement.

Complicating the negotiations, Mr. Cullen, who helped Mr. Pence navigate the Russia investigation without being called as a witness, will have to step aside as Mr. Pence’s lawyer because he will become a top adviser to Virginia’s incoming governor, Glenn Youngkin, when Mr. Youngkin is sworn in this week.

In the absence of Mr. Pence’s cooperation, the committee is trying to learn how Mr. Pence handled the pressure from Mr. Trump. Last fall, the committee interviewed J. Michael Luttig, a former federal judge who found himself coming to Mr. Pence’s aid in the two days leading up to the Jan. 6 attack. Mr. Luttig, in response to a request from Mr. Cullen, a longtime friend, put out a statement that said Mr. Pence had no ability to stop the certification of the election, which Mr. Pence ultimately used as political and legal cover in his decision to buck Mr. Trump.

In their questioning, committee investigators asked Mr. Luttig if Mr. Pence was wavering about what to do in the two days before Jan. 6. Mr. Luttig told the committee that he thought Mr. Pence had decided what to do.

In recent weeks, the committee has moved to question Mr. Pence’s former chief of staff, Marc Short, and his former chief counsel, Greg Jacob, who are considered the two key Pence witnesses.

Mr. Short and Mr. Jacob were both closely involved in Mr. Pence’s consideration of whether to go along with Mr. Trump’s assertions that he could act to block the certification of the Electoral College results. Three days before the Jan. 6 riot, the two men met with Mr. Eastman, a lawyer then advising Mr. Trump, about Mr. Eastman’s memo setting out a case for why Mr. Pence had the power to hold off the certification.

Media reports and a statement from at least one committee member have given the impression that Mr. Pence’s team has provided significant cooperation, and some of his former aides have spoken with the committee. But Mr. Short, arguably the most important witness from the team, publicly attacked the panel’s credibility three weeks ago.

“I can’t have a lot of confidence that this committee is going to provide some sort of impartial analysis,” Mr. Short said on Fox News. “I think that when the Democrats rejected the people that Kevin McCarthy put forward to make it a more bipartisan commission, I think it went down more of a political show-trial path.”

Although Mr. Short was subpoenaed by the committee, he has yet to testify and has refused to commit to cooperating with it. His lawyer, Emmet Flood; committee lawyers; White House lawyers; and the National Archives are negotiating over what topics Mr. Short can discuss and whether any are covered by executive privilege.

If Mr. Pence rebuffs the panel’s request to testify voluntarily, it will be forced to decide whether to subpoena a former vice president, a move that Congress is believed to have not taken since it subpoenaed John Tyler, a former president and vice president, in 1846.