As Mr. Trump attempts to subvert the election to remain in power, Ms. Trump, three allies said, has been telling associates she is considering a run for Senate in 2022, in what is expected to be a competitive race for the first open Senate seat in a very swingy swing state in a generation. Senator Richard Burr, an unobtrusive Republican legislator who was thrust into the spotlight as chairman of a committee investigating the president’s ties to Russia, has said he will retire at the end of his term. Despite expanded turnout in rural areas, Mr. Trump won North Carolina by a smaller margin than he did four years ago, just 1.3 percentage points, a sign that overall the state is trending blue and that the race for the Senate seat will be tightly contested by both parties in the first post-Donald Trump election.

But not, perhaps, an entirely post-Trump election, if Ms. Trump proceeds.

Ms. Trump, 38, a former personal trainer and television producer for Inside Edition, wed Eric Trump at the family’s Mar-a-Lago estate in 2014 and worked as a senior adviser on the 2020 Trump campaign. Now, the daughter-in-law whom Mr. Trump had often joked to donors that he “couldn’t pick out of a lineup” is floating herself as the first test of the enduring power of the Trump name.