Conspiracies surrounding Mr. Sanders’s political fortunes have been a particular fixation for Mr. Trump, dating back four years. During the 2016 campaign, Mr. Trump circulated the false and unsourced claim that an “analysis” — he did not say who wrote it or where it was published — concluded that Mr. Sanders would have won the Democratic nomination if not for superdelegates, the party leaders and officials who were not bound to vote for the winner of their states’ primaries or caucuses.

At the time, Mr. Trump and his advisers realized the potential political benefit in lobbing these kinds of accusations. Their campaign, which relied heavily on depressing Democratic turnout as a way to win battleground states like Florida and Michigan, stood to gain by fanning the flames of the rivalry between Mr. Sanders and Hillary Clinton and dredging up the bitterness that many Sanders supporters felt over their loss.

Even after winning the election, Mr. Trump continued to claim that Mrs. Clinton had somehow robbed Mr. Sanders of victory. When Donna Brazile, the former Democratic National Committee chairwoman, released a memoir in 2017, Mr. Trump inaccurately said the book showed that Mrs. Clinton “bought the DNC & then stole the Democratic Primary” from Mr. Sanders.

In an interview, Mr. Fleischer said he had not seen the idea about the timing of the impeachment trial anywhere else and had not consulted with anyone when he first pitched it on Twitter. “I just do my best to realistically assess what’s happening in Washington,” he said. Mr. Fleischer said he believed that Ms. Pelosi does not think Mr. Sanders can beat Mr. Trump in November, and that “she has one big thing on her mind: that’s winning the White House.”

He said his tweet took off because “if it has merit, it starts to gather momentum.”

“If it has no merit, it’s just another tweet,” he added.

Republican staff members on Capitol Hill said the theory gained traction because of a broader narrative — pushed by Mr. Sanders’s own supporters — that Mr. Sanders was generally getting a raw deal from the mainstream news media and other candidates in the race.

In a statement on Friday, in response to a question from The New York Times about the president’s conspiracy tweet, Mr. Sanders denounced the theories. “Let’s be clear about who is rigging what: It is Donald Trump’s action to use the power of the federal government for his own political benefit that is the cause of the impeachment trial,” he said. “His transparent attempts to divide Democrats will not work, and we are going to unite to sweep him out of the White House in November.”