All indications remain, however, that Mr. Cohen has proved to be a valuable witness for federal prosecutors working for the special counsel, Robert S. Mueller III, who have vouched for his truthfulness, and the Southern District of New York, who appear to have built a case at least in part around his accounts.

In his public testimony before the House Oversight Committee last week, Mr. Cohen emphatically denied that he had ever wanted a White House job or a presidential pardon.

“I did not want to go to the White House,” Mr. Cohen asserted under questioning.

But while he was still seated at the table in the hearing room, one of the president’s sons, Eric Trump, took to Twitter to dispute one of the claims.

“Michael was lobbying EVERYONE to be ‘Chief of Staff,’” he wrote. “It was the biggest joke in the campaign and around the office. Did he just perjure himself again?” Other members of the 2016 Trump campaign backed him up.

Mr. Cohen was just as adamant that he had not sought a pardon from Mr. Trump.

“I have never asked for, nor would I accept, a pardon from Mr. Trump,” he said at the hearing.

A week after that testimony, though, Mr. Cohen’s lawyer, Lanny J. Davis, told The Wall Street Journal that Mr. Cohen had directed his former lawyer, Stephen Ryan, to inquire about the possibility of a pardon with Mr. Trump’s lawyers.

Mr. Davis maintained that there was no discrepancy between that statement and Mr. Cohen’s testimony, insisting that his client had given those instructions only after the Trump team had “dangled” a possible pardon while they were all part of a shared defense. Mr. Davis said that Mr. Cohen was referring in his testimony to the period after early July 2018, when the president’s former fixer split from his old boss.