Mr. Giuliani said that he had spoken with the president before and after his interview on Fox News, and that Mr. Trump and other lawyers on the team were aware of what he would say. On Thursday, Mr. Trump said that he paid a monthly retainer to Mr. Cohen, suggesting, as Mr. Giuliani had, that the payment to Ms. Clifford was not a campaign contribution.
The president has repeatedly denied that he had an affair with Ms. Clifford, who has described having intimate contact with Mr. Trump before he became president.
Mr. Giuliani’s comments are also in direct contrast to what Mr. Cohen has been saying for months — that he used his own money to pay Ms. Clifford, whose stage name is Stormy Daniels. Mr. Cohen is under investigation by the F.B.I., which raided his home and office last month and seized documents that included information about the payment to Ms. Clifford.
“They funneled through a law firm, and the president repaid it,” Mr. Giuliani told Sean Hannity, the Fox News host. After Mr. Hannity asked for clarification, Mr. Giuliani insisted: “That was money that was paid by his lawyer. The president reimbursed that over the period of several months.”
The source of the $130,000 payment is at the center of several legal disputes involving Mr. Trump, Mr. Cohen and Ms. Clifford. That includes whether the payment to Ms. Clifford was in effect a contribution to Mr. Trump’s campaign aimed at preventing a negative article from surfacing just before Election Day.
Mr. Giuliani said that Mr. Cohen had “settled several problems for” Mr. Trump, and that the payment related to them. Another person familiar with the payment confirmed that that was the mechanism used to repay Mr. Cohen.
Mr. Giuliani said that he was “not clear that” Mr. Trump was aware of the payment to Ms. Clifford at the time it was made, and he said that his understanding was that the president did not learn about the payment to her until recently.
“I don’t think he did” know “until now,” Mr. Giuliani said.
The comments on Fox sent a jolt through Washington and New York, including the legal teams working on behalf of the president, Mr. Cohen and Ms. Clifford, who has sued Mr. Cohen in an attempt to be released from the nondisclosure agreement that accompanied the $130,000 payment in October 2016.
Michael Avenatti, Ms. Clifford’s lawyer, said Wednesday night on Twitter that Mr. Giuliani’s comments amounted to an admission that the president had lied to the American people about whether he was aware of the hush payment.
“Mr. Trump stood on AF1 and blatantly lied,” Mr. Avenatti wrote. “This followed the lies told by others close to him, including Mr. Cohen. This should never be acceptable in our America. We will not rest until justice is served.”
In an interview, Mr. Avenatti said a key issue going forward would be how Mr. Trump accounted for any such payment and whether it was hidden in a way that could violate anti-money laundering statutes.
Mr. Giuliani’s statement to Mr. Hannity was the latest of several contradictory narratives from Mr. Trump’s representatives about the deal to pay Ms. Clifford for her silence during the campaign, each of which have had potential implications for violating campaign finance disclosure laws.
Mr. Cohen had previously provided a detailed account of how he had paid Ms. Clifford $130,000 out of his own pocket, drawing down on a home loan to do so. That opened Mr. Cohen and the Trump campaign to charges that he violated the law by exceeding the amount that an individual can donate to a presidential campaign.
If Mr. Trump reimbursed Mr. Cohen for the payment, that could help Mr. Cohen and the campaign avoid prosecution for violating campaign contribution limits because a candidate is allowed to contribute an unlimited amount to his own campaign.
According to a person close to the president, Mr. Trump reimbursed Mr. Cohen for the payment, a determination made in recent weeks by the president’s lawyers after examining banking records. The lawyers determined that campaign funds had not been used for the payment, the person said.
The lawyers also concluded that the funds did not come from a third party or the Trump Organization, the person said. While the disclosure may be embarrassing for the president, his lawyers believe the payment did not violate campaign finance laws, the person said.
Another person familiar with the payment to Mr. Cohen from Mr. Trump said the money was reimbursed in a series of payments, not in a lump sum.
But such a payment from the candidate — even if it was made through a lawyer — would have to be disclosed to the Federal Election Commission as an in-kind contribution to the campaign and as an expenditure by the campaign, if it was for the purpose of influencing the election.
Mr. Trump’s campaign did not disclose the reimbursement to Mr. Cohen on its commission reports.
The crucial question in determining whether the reimbursement to Mr. Cohen violated campaign finance laws might be whether the payment was specifically intended to help Mr. Trump’s campaign.
Paul S. Ryan, an official at the government watchdog group Common Cause, argued that “all the facts indicate that the payment was to influence the election.”
Mr. Ryan asserted that Mr. Giuliani’s admission could allow prosecutors to make the case that Mr. Trump “knowingly caused his campaign committee to file an incomplete disclosure report with the F.E.C.”
“Until tonight, it would have been tough to prove that because Donald Trump had denied knowing about the payment,” Mr. Ryan said. “But his reimbursement amounts to knowledge.”
While most violations of the Federal Election Campaign Act’s disclosure provisions are misdemeanors, a knowing and willful violation could be a felony.
But Mr. Giuliani’s comments could cloud Mr. Avenatti’s legal argument that his client should be formally released from the hush agreement she signed. When the president told reporters he knew nothing about the deal, he helped Ms. Clifford’s case that the hush agreement was not binding if Mr. Trump did not know about it and did not sign it.
Though far from bringing clarity to the question, Mr. Giuliani obscured it further — now two lawyers for the president are providing two very different versions of events.