But a senior White House official said that Donald F. McGahn II, the president’s chief counsel, was working on Saturday to secure access to what Mr. McGahn believed was 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. If one does, it would be highly unusual for a White House to order the Justice Department to turn over such an investigative document, given the traditional independence of law enforcement matters.

It has been widely reported that there is a federal investigation, which began during the 2016 presidential campaign, into links between Trump associates and the Russians. That issue has dogged Mr. Trump for months.