WASHINGTON — Rudolph W. Giuliani, who helped lead President Donald J. Trump’s effort to overturn the results of the 2020 election as his personal lawyer, on Thursday abruptly pulled out of a scheduled Friday interview with the House committee investigating the Jan. 6 attack on the Capitol after the panel refused to let him record the session.

Mr. Giuliani has been negotiating with the panel about testifying for months, and had finally reached an agreement to speak about matters other than his conversations with Mr. Trump or any other topic he believes is covered by attorney-client privilege, said his lawyer, Robert J. Costello.

Mr. Giuliani’s sudden withdrawal threatens what could have been a major breakthrough for the investigation. His testimony could have included details about interactions with members of Congress and others involved in the plans who were not Mr. Giuliani’s clients, Mr. Costello said. And with Mr. Giuliani under a subpoena to testify, the standoff raises the specter of yet another protracted legal battle between the committee and a former Trump aide.

The impasse began when Mr. Costello told the committee on Thursday that Mr. Giuliani intended to record the interview on video. When the panel’s lawyers refused to allow him to do so, he canceled the meeting, Mr. Costello said.

“He’s willing to talk about anything that is not privileged,” Mr. Costello said. “The only sticking point we’ve had is recording the interview. Now, that tells me they’re more interested in keeping things secret than getting to the so-called truth. If you’re interested in the person’s testimony, why would you not agree to this?”

Mr. Costello said that Mr. Giuliani’s testimony was not yet off the table, and that he would continue to negotiate with the panel’s lawyers. He said that Mr. Giuliani, a former New York mayor, “simply doesn’t trust” members of the committee, specifically Representative Adam B. Schiff, Democrat of California, and believes that they will selectively edit his testimony.

“If they changed their mind and they said, ‘Listen, we’ll jointly record the interview,’ then we would participate,” Mr. Costello said.

Tim Mulvey, a spokesman for the committee, said the panel would consider enforcement actions against Mr. Giuliani if he does not change course and comply with the committee’s subpoena.

“Mr. Giuliani had agreed to participate in a transcribed interview with the select committee. Today, he informed committee investigators that he wouldn’t show up unless he was permitted to record the interview, which was never an agreed-upon condition,” Mr. Mulvey said. “Mr. Giuliani is an important witness to the conspiracy to overthrow the government, and he remains under subpoena. If he refuses to comply, the committee will consider all enforcement options.”

The committee has interviewed more than 970 witnesses and has recommended criminal contempt of Congress charges against four of Mr. Trump’s closest allies, who have refused to fully cooperate.

Mr. Trump’s final chief of staff, Mark Meadows, has been referred to the Justice Department for possible criminal charges after refusing an interview with the committee. Another former aide, Stephen K. Bannon, was indicted in November after refusing to provide information to congressional investigators.

Last month, the House voted to recommend criminal contempt of Congress charges against Peter Navarro and Dan Scavino Jr., two other close allies of Mr. Trump, after the pair defied subpoenas from the committee.

The Justice Department has yet to act on the referrals for Mr. Meadows, Mr. Navarro and Mr. Scavino.

As a key figure in Mr. Trump’s attempts to stave off electoral defeat, Mr. Giuliani would be in a position to tell investigators about a series of extraordinary measures undertaken last fall and winter in a bid to maintain the losing president’s grip on power.

Among those efforts was a scheme to disrupt the normal workings of the Electoral College by persuading lawmakers in contested swing states to draw up alternate slates of electors showing Mr. Trump was victorious in states that were actually won by President Biden.

Mr. Giuliani was also instrumental in vetting a plan to use the Department of Homeland Security to seize voting machines and examine the data housed inside them for supposed evidence of fraud. At Mr. Trump’s direction, Mr. Giuliani asked a top homeland security official if the department could legally take control of the machines — a notion the official shot down. Mr. Giuliani later opposed an even more explosive proposal to have the military seize the machines.

Mr. Giuliani was subpoenaed with other members of a legal team that billed itself as an “elite strike force” and pursued a set of lawsuits on behalf of Mr. Trump in which they promulgated conspiracy theories and made unsubstantiated claims of fraud in the election.

The committee’s subpoena sought all documents that Mr. Giuliani had detailing the pressure campaign that he and other Trump allies initiated targeting state officials, the seizure of voting machines, contact with members of Congress, any evidence to support the conspiracy theories he pushed and any arrangements for his fees.

On Jan. 6, speaking to a crowd of Trump supporters before a pro-Trump mob attacked the Capitol, Mr. Giuliani called for “trial by combat.” Later, after the building was under siege, both he and Mr. Trump called lawmakers in an attempt to delay the certification of Mr. Biden’s victory.