It was a shocking double victory for the rioters who attacked the Capitol on Jan. 6, 2021, when President Trump granted them clemency for their crimes on his first day back in the White House and then, in the months that followed, allowed his Justice Department to purge many of the federal agents and prosecutors who sought to hold them accountable.

But even though the president has given them their freedom and has taken steps toward satisfying their desire for retribution, they are asking for more: In the past several weeks, the rioters and their lawyers have made a concerted effort to push the Trump administration into paying them restitution for what they believe to be their unfair prosecutions.

On Thursday, one of the lawyers, Mark McCloskey, said during a public meeting on social media that he had recently met with top officials at the Justice Department and pitched them on a plan to create a special panel that would dole out financial damages to the rioters — much like the arrangement of a special master to award money to the victims of the Sept. 11, 2001, terrorist attacks.

The panel, which Mr. McCloskey called a “voluntary nonjudicial resolution committee,” would consider rioters’ cases individually, he said, then assign them sums according to harms they had purportedly suffered at the hands of the federal government.

Mr. McCloskey said that he wanted the panel to be overseen by Jeanine Pirro, who runs the federal prosecutors’ office in Washington that took the lead in filing charges against nearly 1,600 rioters who joined in the Capitol attack.

“The only thing I can do as your lawyer,” he told the rioters who were at the online meeting, “is to turn your losses into dollar bills.”