“The only plot is to interfere in the election by the Russians,” he added.

Since returning from a widely criticized meeting with President Vladimir V. Putin of Russia in Helsinki, Finland, Mr. Trump has fixated on election interference, but he has been ready to place blame not on the Russians but on the Democrats. On the plane ride home from Finland, Mr. Trump asked advisers once again about the Democratic National Committee server that was hacked — he had raised the server issue while standing next to Mr. Putin — and why cyberintruders had not penetrated Republican National Committee systems.

Supporters of Mr. Trump, including Republicans on the House Intelligence Committee, have seized on the fact that the F.B.I., in making the case to judges that Mr. Page might be a Russian agent, used some claims included in a notorious Democratic-funded dossier compiled by Christopher Steele, a former British intelligence agent.

Republicans’ criticism has centered on the fact that the F.B.I. used material from the dossier without telling the court that the Democratic National Committee and the Clinton campaign, by name, had funded the research.

“I don’t have an issue with looking into people that have cozy relationships with Russia,” Mr. Gowdy said. “I do have an issue when you rely on unvetted political opposition research.”

But the application shows that the F.B.I. acknowledged to the court that it believed that the person who hired Mr. Steele was looking for information to discredit Mr. Trump’s campaign, later emphasizing that regardless of Mr. Steele’s reason for conducting research into Mr. Trump’s ties to Russia, it believed his reporting about Mr. Page was credible.

The application also shows that the Justice Department’s general practice in surveillance applications was not to specifically name Americans or American organizations. For example, it referred to Mr. Trump not by name but as “Candidate #1,” despite noting in renewal applications that this person had since become president.

Mr. Trump and some of his supporters nevertheless claimed vindication. In another tweet on Sunday, the president quoted a Fox News commentator, Andrew McCarthy, as saying: “I said this could never happen. This is so bad that they should be looking at the judges who signed off on this stuff, not just the people who gave it. It is so bad it screams out at you.”