During the 2016 campaign, the federal authorities began an investigation into links between Trump associates and the Russian government, an issue that continues to dog Mr. Trump. His aides declined to clarify on Saturday whether the president’s allegations were based on briefings from intelligence or law enforcement officials — which could mean that Mr. Trump was revealing previously unknown details about the investigation — or on something else, like a news report.

But a senior White House official said that Donald F. McGahn II, the president’s chief counsel, was working to secure access to what Mr. McGahn believed to be an order issued by the Foreign Intelligence Surveillance Court authorizing some form of surveillance related to Mr. Trump and his associates.

The official offered no evidence to support the notion that such an order exists. It would be a highly unusual breach of the Justice Department’s traditional independence on law enforcement matters for the White House to order it to turn over such an investigative document.

Any request for information from a top White House official about a continuing investigation would be a stunning departure from protocols intended to insulate the F.B.I. from political pressure. It would be even more surprising for the White House to seek information about a case directly involving the president or his advisers, as does the case involving the Russia contacts.

After the White House received heavy criticism for the suggestion that Mr. McGahn would breach Justice Department independence, a different administration official said that the earlier statements about his efforts had been overstated. The official said the counsel’s office was looking at whether there was any legal possibility of gleaning information without impeding or interfering with an investigation. The counsel’s office does not know whether an investigation exists, the official said.