The memo in fact undermined Republicans’ efforts to cast doubt on the roots of the investigation by confirming that the inquiry was already underway when law enforcement officials obtained a warrant from a secret intelligence court to conduct surveillance on Mr. Page.

The Republican document, which Democrats dismissed as containing cherry-picked information and focusing on an obscure figure in the Trump campaign, confirms that a primary factor in the opening of the investigation in July 2016 was initial contacts between a former Trump foreign policy adviser, George Papadopoulos, and Russian intermediaries.

But Republicans say the memo raises serious questions. They say the F.B.I. should have told the court that the source of some of the information on which it was basing its request for a warrant involving Mr. Page had been paid for by Democrats.

Returning to Twitter on Saturday night, Mr. Trump quoted from a Wall Street Journal editorial that claimed the F.B.I. had become “a tool of anti-Trump political actors.”