WASHINGTON — On the day that President Grover Cleveland left the White House after losing his bid for re-election, his wife, Frances, told the staff to take care of the mansion. “I want to find everything just as it is now when we come back again,” she said. “We are coming back just four years from today.” And they did.

Nearly 128 years after the Clevelands’ triumphal comeback, President Trump is signaling that he may try to become only the second president in American history to win another term after being defeated. Even as he insists that he did not lose the 2020 election, Mr. Trump has been laying plans to run again in 2024 with a kickoff as early as this month or possibly on his successor’s Inauguration Day.

How serious he is remains to be seen. Many allies believe his talk of another run in 2024, when he will be 78 years old, is more about maintaining his relevance, enabling him to raise funds, soothing his wounded pride and trying to shed the label of loser. But even if it is only for show, Mr. Trump’s talk of a 2024 race has already frozen the Republican field and could delay the emergence of a new generation of leaders while keeping the party tethered to the politically polarizing Mr. Trump for months or years to come.

The outgoing president hinted at his ambitions to a crowd of Republican supporters at a White House Christmas party on Tuesday evening in his most overt semipublic comments since losing the Nov. 3 election that he refuses to concede. “It’s been an amazing four years,” he told guests in remarks posted online by a member of the Republican National Committee. “We’re trying to do another four years. Otherwise, I’ll see you in four years.”

If so, it would be only the latest effort by Mr. Trump to defy history and do what few if any have done before. At least a half-dozen former presidents have sought to come back in the past, but none other than Cleveland succeeded, in some cases tarnishing their reputations in the losing effort.