Despite Mr. Trump’s claims, the Republican memo did nothing to clear him of either collaborating with the Russians or obstructing justice — the two lines of inquiry being pursued by the special counsel, Robert S. Mueller III. The memo undermined an effort by some Republicans to cast doubt on the roots of the investigation by confirming that contacts between another former Trump foreign policy adviser, George Papadopoulos, and Russian intermediaries — not the material from Mr. Steele — were the primary factor in the opening of the investigation in July 2016.

Nevertheless, Democrats have denounced the document as a tactic to undermine the investigation and to protect Mr. Trump, and they have said it is riddled with errors and omissions.

Specifically, the Democratic memo is said to contend that the F.B.I. was more forthcoming with the surveillance court than Republicans had claimed. People familiar with the Democratic document said that it reveals that while the F.B.I. did not name the Democratic National Committee or Mrs. Clinton’s campaign, the bureau disclosed to the court that the information it had received from Mr. Steele was politically motivated.

Confronted on Monday with Democratic claims that such a disclosure was included in the application, Mr. Nunes conceded on “Fox & Friends” that there had been a “footnote” to that effect. But he called its inclusion a “far cry from letting the American people know that the Democrats and the Hillary campaign paid for dirt that the F.B.I. then used to get a warrant.”

Democrats also contest Republican claims that Andrew G. McCabe, the deputy director of the F.B.I. at the time, had testified before the Intelligence Committee late last year that the agency would not have sought a wiretap of Mr. Page without Mr. Steele’s dossier of information.

The F.B.I. suspected that Mr. Page, a former Moscow-based investment banker who was under investigation once before, was acting as a Russian agent.

The New York Times filed a motion on Monday asking the Foreign Intelligence Surveillance Court to unseal all materials related to the wiretap of Mr. Page, including the F.B.I.’s application for the warrant and other court documents. Since Congress enacted the Foreign Intelligence Surveillance Act in 1978, no such wiretapping application materials have been made public.