WASHINGTON — President Trump said on Wednesday that Susan E. Rice, the former national security adviser, may have committed a crime by seeking to learn the identities of Trump associates swept up in surveillance of foreign officials by United States spy agencies, repeating an assertion his allies in the news media have been making since last week.

Mr. Trump gave no evidence to support his claim, and current and former intelligence officials from both Republican and Democratic administrations have said they do not believe Ms. Rice’s actions were unusual or unlawful. The president repeatedly rebuffed attempts by two New York Times reporters to learn more about what led him to the conclusion, saying he would talk more about it “at the right time.”

The allegation by a sitting president was a remarkable escalation — and, his critics say, the latest effort to change the story at a time when his nascent administration has been consumed by questions about any role his associates may have played in a Russian campaign to disrupt last year’s presidential election.

Since March 4, when Mr. Trump posted on Twitter that President Barack Obama had “wiretapped” him at Trump Tower during the campaign, the president and his allies have repeatedly sought evidence trying to corroborate that claim, despite flat denials from James B. Comey, the director of the F.B.I., and other senior intelligence officials.