Representative Jamie Raskin, Democrat of Maryland and a member of the Jan. 6 committee, said a huge difference between the first two years of Democratic control of the House and now is President Biden, who has aligned his White House counsel and his Justice Department with the House’s oversight efforts.

“The legislative and executive branches are completely in agreement with each other, that this material is not privileged and needs to be turned over to Congress,” he said. “So I think things have been moving much more quickly.”

Representative Adam B. Schiff, Democrat of California and another committee member, noted that the House’s two contempt referrals, against Mr. Bannon and Mr. Meadows, were criminal cases. If the Justice Department decides to prosecute Mr. Meadows, as it did Mr. Bannon, both men would face the prospects of jail time and fines.

“And that would be true irrespective of who’s controlling the Congress,” Mr. Schiff said.