“If this was a first-time candidate without a public reputation, then it would be harder to argue that it wasn’t an expenditure to influence an election,” said Charles Spies, a Republican election lawyer who worked in support of one of Mr. Trump’s rivals, the former Florida governor Jeb Bush, in the 2016 Republican primary. “But Donald Trump has a long record of aggressively defending his reputation from attacks.”

That track record could prove less compelling if the authorities obtain evidence that Mr. Cohen privately discussed the payment in the context of Mr. Trump’s campaign. Last month, federal law enforcement officials in New York raided Mr. Cohen’s office and hotel room, carting away numerous documents and electronic devices that are now the subject of a fight over attorney-client privilege. One of the things they are apparently investigating is the payment to Ms. Clifford.

If the payment was a loan, is Cohen out of trouble?

Not if the loan was intended to influence the election. Campaigns routinely take out large loans from banks when they’re running short of cash ahead of elections. But federal election laws require that those loans come from banks as routine business transactions. Personal loans count as contributions that are still legally capped at the individual contribution limit — even if they are later repaid in full, according to the Federal Election Commission website.

When would this have had to be reported, and by whom?

If the payment were to be deemed campaign-related, the Trump campaign should have disclosed it in its periodic filings with the F.E.C., as soon as Mr. Trump or his campaign learned that Mr. Cohen had made it. If Mr. Trump or his campaign only discovered the payment after the fact, they should have amended their previous filings to reflect the expenditure, and any reimbursements to Mr. Cohen. The Wall Street Journal first reported the $130,000 payment to Ms. Clifford in January, and Mr. Trump has not moved to amend his disclosures.

In an interview with The Times, Mr. Giuliani was vague about key questions concerning what Mr. Trump knew and when he knew it, saying that Mr. Trump “did know that there was a form of reimbursements” to Mr. Cohen over the course of 2017, but maintaining that the president did not know what it was for specifically. Mr. Giuliani also said that some executives at the Trump Organization “knew about the fact that Cohen believes money was owed to him — I don’t know when that came up.”