WASHINGTON — The House committee investigating the Jan. 6 Capitol riot moved on Wednesday to hold Jeffrey Clark in criminal contempt of Congress for refusing to cooperate with its inquiry, but agreed to delay a House vote on the matter as the former Justice Department lawyer made an 11th-hour offer to be interviewed again.

The panel voted unanimously to recommend charging Mr. Clark, who had pressed his colleagues at the Justice Department to pursue President Donald J. Trump’s election fraud claims, after he refused to answer any questions or produce any documents at a deposition with its investigators last month.

The vote paved the way for the full House to move quickly to call on the Justice Department to prosecute Mr. Clark for his refusal to cooperate with the panel’s subpoena. But shortly before the committee met to approve it, Mr. Clark had requested a delay of the proceedings, offering to sit down with the panel again.

Representative Bennie Thompson, Democrat of Mississippi and the committee’s chairman, said it would move forward with the contempt referral anyway, calling Mr. Clark’s appeal “a last-ditch attempt to delay.”

“The select committee has no desire to be placed in this situation, but Mr. Clark has left us no other choice,” Mr. Thompson added. “He chose this path. He knew what consequences he might face if he did so. This committee and this House must insist on accountability in the face of that sort of defiance.”

But he announced that the panel had set another deposition for Mr. Clark on Saturday, and that it would not seek a House vote on the contempt charge until investigators had determined whether he was willing to cooperate.

It was not immediately clear to what extent Mr. Clark planned to do so. In a letter to the panel on Tuesday, he offered a new rationale for refusing to answer questions, asserting his Fifth Amendment right against self-incrimination. Mr. Thompson said Mr. Clark would be permitted to invoke that right “on a question-by-question basis” during the upcoming interview.

The last time the committee’s investigators sat down with Mr. Clark, they had a long list of questions about his role in trying to help Mr. Trump invalidate his 2020 election defeat.

They wanted to ask Mr. Clark about a national intelligence briefing he had sought about a wild theory that China could hack voting machines through thermostats. They planned to press him about a letter he had proposed writing to legislative officials in Georgia, urging them to put forward an alternative slate of electors for Mr. Trump, instead of President Biden, who had won the state. And they wanted to dig into any conversations he might have had with a group of Mr. Trump’s allies who had gathered at a Washington, D.C., hotel in the days before the riot to plan the effort to overturn the election.

Mr. Clark offered no answers.

“We will not be answering any questions or producing any documents,” Mr. Clark’s lawyer, Harry W. MacDougald, said flatly.

The vote was the second such confrontation between the committee and an ally of Mr. Trump since Congress began investigating the circumstances surrounding the Capitol riot, including the former president’s attempts to subvert the election.

The House voted in October to recommend that another of Mr. Trump’s associates, Stephen K. Bannon, be charged with criminal contempt of Congress for stonewalling the inquiry. A federal grand jury subsequently indicted him on two counts that could carry a total of up to two years behind bars.

A third recalcitrant witness, Mark Meadows, a White House chief of staff under Mr. Trump, reached an agreement with the committee on Tuesday to provide documents and appear voluntarily for a deposition. It is a notable reversal for a crucial witness in the inquiry, though it is not clear how much information he will be willing to provide.

The committee has interviewed more than 200 witnesses and issued 45 subpoenas. On Tuesday, the panel heard five hours of closed-door testimony from Georgia’s secretary of state, Brad Raffensperger, a Republican who pushed back against Mr. Trump’s attempts to overturn the election results there.

Some of the panel’s most sought-after witnesses, including Mr. Meadows; Dan Scavino Jr., a former deputy chief of staff; and Kash Patel, a former Pentagon chief of staff, are scheduled to testify, Mr. Thompson said.

Under federal law, any person summoned as a congressional witness who refuses to comply can face a misdemeanor charge that carries a fine of $100 to $100,000 and a jail sentence of one month to one year.

In rebuffing the committee’s October subpoena, Mr. Clark said his conversations with Mr. Trump were protected by attorney-client privilege and the former president’s assertion of executive privilege.

Mr. MacDougald told the committee last month that he and Mr. Clark interpreted executive privilege to cover conversations and documents that did not involve Mr. Trump.

“The privileges that are under the overall umbrella of executive privilege are numerous,” Mr. MacDougald said.

He also cited law enforcement privilege and deliberative process privilege. “There are any number,” he added.

Mr. Trump has filed suit against the committee, seeking to block its access to hundreds of White House documents, though a federal appeals court on Tuesday appeared skeptical of his claim that he has the power to block the panel’s demand for White House records related to the attack on the Capitol.

The Biden administration has declined to assert privilege over the documents, arguing that no such protection should be afforded to material that could shed light on a president’s attempts to undermine a democratic election.