In the memo in the case of Mr. Cohen, prosecutors from the Southern District of New York depicted Mr. Trump, identified only as “Individual-1,” as an accomplice in the hush payments. While Mr. Trump was not charged, the reference echoed Watergate, when President Richard M. Nixon was named an unindicted co-conspirator by a grand jury investigating the cover-up of the break-in at the Democratic headquarters.
“While many Americans who desired a particular outcome to the election knocked on doors, toiled at phone banks or found any number of other legal ways to make their voices heard, Cohen sought to influence the election from the shadows,” the prosecutors wrote.
“He did so by orchestrating secret and illegal payments to silence two women who otherwise would have made public their alleged extramarital affairs with Individual-1,” they continued. “In the process, Cohen deceived the voting public by hiding alleged facts that he believed would have had a substantial effect on the election.”
The exposure on campaign finance laws poses a challenge to Mr. Trump’s legal team, which before now has focused mainly on rebutting allegations of collusion and obstruction while trying to call into question Mr. Mueller’s credibility.
“Until now, you had two different charges, allegations, whatever you want to call them,” Representative Jerrold Nadler of New York, the incoming Democratic chairman of the House Judiciary Committee, said in an interview on Saturday. “One was collusion with the Russians. One was obstruction of justice and all that entails. And now you have a third — that the president was at the center of a massive fraud against the American people.”
The episode recalled a criminal case brought against former Senator John Edwards, Democrat of North Carolina, who while running for president in 2008 sought to cover up an extramarital affair that resulted in pregnancy. He was charged with violating campaign finance laws stemming from money used to hide his pregnant lover, but a trial ended in 2012 with an acquittal on one charge and a mistrial on five others.
Mr. Giuliani pointed to that outcome on Saturday to argue that the president should not be similarly charged.
“The President is not implicated in campaign finance violations because based on Edwards case and others the payments are not campaign contributions,” Mr. Giuliani wrote on Twitter. “No responsible prosecutor would premise a criminal case on a questionable interpretation of the law.”
But Mr. Cohen has pleaded guilty under that interpretation of the law, and even if Mr. Trump cannot be charged while in office, the House could still investigate or even seek to impeach him. The framers of the Constitution specifically envisioned impeachment as a remedy for removing a president who obtained office through corrupt means, and legal scholars have long concluded that the threshold of “high crimes and misdemeanors” does not necessarily require a statutory crime.
If the campaign finance case as laid out by prosecutors is true, Mr. Nadler said, Mr. Trump would be likely to meet the criteria for an impeachable offense, and he said he would instruct his committee to investigate when he takes over in January.
But he added that did not necessarily mean that the committee should vote to impeach Mr. Trump. “Is it serious enough to justify impeachment?” he asked. “That is another question.”
The strategy of Mr. Trump’s lawyers has been predicated on the assurance by senior Justice Department officials that if Mr. Mueller found evidence that the president broke the law, he would not be indicted while in office. But the hush money investigation is being led by a separate office of prosecutors in New York, and far less time has been spent publicly or privately trying to protect Mr. Trump from that inquiry.