Mr. Steele had been retained to investigate Mr. Trump by Fusion GPS, a research firm that had been hired by a law firm working for Democrats. When Mr. Steele decided to take the information he was gathering to the F.B.I. in August and October 2016, according to people familiar with those conversations, he told the bureau’s agents that he was working for interests opposed to Mr. Trump’s campaign. Fusion GPS, which was under contract with the Democrats at the time, paid around $160,000 for Mr. Steele’s research.

It is not clear how much of that context — if any — law enforcement officials conveyed to the court.

Two people familiar with the warrant application said law enforcement officials described Mr. Steele not as a Democratic-funded investigator, but as a reliable F.B.I. source who had previously provided information about corruption in FIFA, the global governing body for soccer.

Stephen Vladeck, a University of Texas law professor who specializes in national security issues, said that disclosing Mr. Steele’s backers could affect a judge’s assessment of his credibility. But Mr. Vladeck cautioned that without seeing the warrant application, it was impossible to say whether that information was just “background noise” or pivotal.

“There is a world of difference between an application that relied solely on this information and an application that relied on this and 22 pieces of independently corroborated information,” he said.

In a transcript of a congressional interview made public on Thursday, Glenn R. Simpson, the founder of Fusion GPS, told the House Intelligence Committee that Mr. Steele decided on his own to go to the F.B.I. and that neither of them had asked permission from their client, nor received instructions from their client to do so.

Republicans have been suggesting for weeks that the Russia investigation traces back to a conspiracy by top F.B.I. officials to sabotage Mr. Trump. They have noted that two officials who were involved in the Russia investigation — Peter Strzok, who led the F.B.I. counterespionage section in 2016, and Lisa Page, an F.B.I. lawyer — exchanged texts indicating that they disliked Mr. Trump and wanted him to lose the election.