Mr. Cohen’s new account of the hotel deal will inevitably be compared not only to the president’s, but also to those of Donald Trump Jr., his eldest son, who testified repeatedly before congressional committees last year about that project and other matters. The complaint states that Mr. Cohen misled Congress about the fact that he had briefed Trump family members about the project. Although the family members were not named, a person familiar with the situation said Mr. Cohen discussed the deal with Donald Trump Jr. and Ivanka Trump.

Testifying before the Senate Judiciary Committee in September 2017, the younger Mr. Trump said that he was only “peripherally aware” of the proposed venture to build a new Moscow hotel bearing the Trump name. Most of what he knew about it, he said, he had learned recently while preparing to testify. He told the committee that Mr. Cohen pursued the project in 2015 and told the House Intelligence Committee that the deal fell dormant as of June 2016 — an accurate date, according to Thursday’s court documents.

Lawyers for Ms. Trump and her brother declined to comment on their interactions with Mr. Cohen about the project. But people close to the Trump family said that while emails indicate that both of them were aware of Mr. Cohen’s efforts to get it off the ground in 2015, their involvement appears to end in January 2016. If Mr. Cohen tried to continue with the project after that, these people added, they did not know.

The president’s critics will also inevitably seek to learn whether he knew that Mr. Cohen, then still a trusted aide, falsely testified to Congress last year. As they continue to do, Mr. Trump’s lawyers then closely monitored what witnesses like Mr. Cohen were telling the authorities through joint defense agreements with their lawyers.

But even if the president knew that Mr. Cohen misled Congress, legal experts said, he is not in legal jeopardy as long as he did not ask Mr. Cohen to lie. And there is no allegation that he did so.

The latest complaint, on its face, seems less worrisome for the president than the previous one lodged against Mr. Cohen in August, said Chuck Rosenberg, a former United States attorney and senior F.B.I. official. In that case, Mr. Cohen directly implicated Mr. Trump in a crime, saying he instructed him to pay money to two women to cover up a potential sex scandal that he feared could endanger his presidential candidacy. Mr. Cohen pleaded guilty to campaign finance violations and a host of other charges for which he has yet to be sentenced.

The president lashed out fiercely at Mr. Cohen on Thursday, calling him a “weak” prevaricator who concocted a false tale about the hotel deal in hopes of winning a lesser punishment. That charge conflicts with Mr. Giuliani’s assertion that Mr. Cohen and Mr. Trump have now given prosecutors much the same account.

Asked how he reconciled the seeming contradiction, Mr. Giuliani blamed Mr. Cohen.

“He has so many different versions of the same stories, so by definition he is a liar and we can’t trust him,” Mr. Giuliani said. “Given the fact that he’s a liar, I can’t tell you what he’s lying about.”