“The $35,000 is an indication of the quality of that evidence, and it both shows the extent of Trump’s leading role and now leaves little doubt that he faces criminal prosecution after he leaves office for the same offenses for which Cohen will serve time,” said Robert F. Bauer, a law professor at New York University and former White House counsel for President Barack Obama.

Indeed, some people close to Mr. Trump have privately predicted that he will ultimately choose to seek a second term in part because of his legal exposure if he is not president. While there is no legal consensus on the matter, Justice Department policy says that a president cannot be indicted while in office.

The checks to Mr. Cohen were sent on a more or less monthly basis throughout 2017, Mr. Trump’s first year in office. Mr. Cohen provided two checks to the House committee last week and his lawyer, Lanny J. Davis, provided the additional six checks to The Times this week. Mr. Cohen’s team said it was searching for three others.

Of the eight checks now available, seven were for $35,000 and another was for $70,000 to cover two months’ worth of payments. Six were signed by Mr. Trump himself while he was president and the other two were signed by his eldest son, Donald Trump Jr., and his company’s chief financial officer, Allen Weisselberg.

Altogether, Mr. Trump or his trust paid Mr. Cohen $420,000, according to federal prosecutors. Of that, $130,000 was to reimburse payments made shortly before the 2016 election to Ms. Daniels, whose legal name is Stephanie Clifford, so she would not tell her story. Another $50,000 was for Mr. Cohen’s effort to manipulate online polls to inflate Mr. Trump’s reputation as a businessman.

That $180,000 was then “grossed up” with another $180,000 to offset taxes that Mr. Cohen would have to pay on the original money since it was being treated as income. Another $60,000 was added as a “bonus,” prosecutors have said.

Mr. Trump has offered conflicting accounts about the matter. Speaking to reporters on Air Force One in April 2018, he said he did not know about the payment to Ms. Daniels. But a month later, his lawyer, Rudolph W. Giuliani, hoping to refute the suggestion that it might be a campaign finance violation, told Fox News and The Times that the president had in fact reimbursed Mr. Cohen for the payment.