House Republicans late Wednesday canceled plans to begin contempt of Congress proceedings against Christopher A. Wray, the F.B.I. director, after the agency agreed to make available to all members of the Oversight Committee a document containing a years-old unsubstantiated allegation of bribery against President Biden.

The decision was a rare dialing back of a concerted effort by House Republicans to target the federal law enforcement agency as they seek to push accusations of wrongdoing by Mr. Biden.

Mr. Wray’s team allowed Representative James Comer, Republican of Kentucky and chairman of the Oversight Committee, and Representative Jamie Raskin of Maryland, the ranking Democrat, to view a redacted copy of the document in a secure area of the Capitol on Monday and briefed them on it for more than an hour.

But the F.B.I. declined to allow lawmakers to take copies of the document out of the secure facility or to show them an unredacted version that reveals the name of a longtime bureau informant, arguing such an action could compromise its source and scare other informants from cooperating with the bureau’s investigations.

That angered Mr. Comer, who had scheduled a vote on Thursday in his committee on a resolution to hold Mr. Wray in contempt of Congress.

“The F.B.I. has caved,” Mr. Comer said in a statement on Wednesday night, “and is now allowing all members of the Oversight and Accountability Committee to review this unclassified record that memorializes a confidential human source’s conversations with a foreign national who claimed to have bribed then-Vice President Joe Biden.”

Mr. Raskin described the allegations after the briefing this week as “hearsay.”

“In the spirit of good faith, the F.B.I. has offered Chairman Comer yet further accommodations in response to his subpoena, including to allow all Oversight Committee members to review in camera the secondhand allegations by Ukrainian individuals reported in the tip sheet,” Mr. Raskin said, adding: “Holding someone in contempt of Congress is among the most serious actions our Committee can take and it should not be weaponized to undermine the F.B.I.”

A person familiar with the agency’s inner workings said on Wednesday that the F.B.I. had offered to provide a similar briefing to the full Oversight Committee as a way of satisfying a demand from Speaker Kevin McCarthy, who said Wednesday he was hopeful an arrangement could be worked out.

“He needs to show it to every Republican and every Democrat on the committee,” Mr. McCarthy said of Mr. Wray. “If he is willing to do that, then there’s not a need to have contempt.”

Since taking over the House, Republicans have vowed to use their subpoena power to investigate Mr. Biden’s son Hunter and brother James in hopes of tying their international business dealings to the president. To date, they have failed to prove that the elder Mr. Biden took payments from foreign business or that he took any corrupt action.

Their efforts to target the F.B.I. and the Justice Department have come after those agencies launched inquires into the conduct of former President Donald J. Trump, who is seeking the Republican presidential nomination and has been informed that he is a target of a federal investigation into his handling of classified documents after he left office.

Republicans have long alleged that Hunter Biden used his seat on the board of the Ukrainian energy company Burisma — for which he was paid substantial sums — to influence his father. A career State Department official raised concerns with a senior White House official in 2015 about the situation.

The attention on Hunter Biden from the right reached a high point in 2020, after Mr. Trump’s personal lawyer Rudolph W. Giuliani began circulating materials about the younger Mr. Biden, including photos and documents detailing drug use and prostitution from a laptop he had abandoned at a Delaware repair shop.

In 2020, Attorney General William P. Barr asked the top federal prosecutor in Pittsburgh, Scott W. Brady, to examine any information Mr. Giuliani had on the Biden family and send anything that might be useful for other prosecutors to them.

One person familiar with the material who insisted on anonymity to discuss it said some of the materials Mr. Brady reviewed were junk and plainly not credible. The bribery allegation against Mr. Biden, which dates to 2020, was never elevated to a preliminary investigation, according to people familiar with the inquiry, but Mr. Brady did forward some information from his work to other prosecutors.

The U.S. attorney in Delaware has conducted a wide-ranging investigation into Hunter Biden and his business dealings, but that inquiry has narrowed significantly and is now focused on potential tax and gun violations.

Mr. Comer has argued his committee, which has hired James Mandolfo, a former federal prosecutor who has experience investigating foreign corruption, needs the document on the elder Biden so congressional investigators “may be able to subpoena relevant bank accounts, interview pertinent witnesses, and corroborate” the informant’s claims.

Mr. Comer said that as part of the agreement the F.B.I. is making two additional documents related to the allegations available to Mr. Comer and Mr. Raskin for review.

Michael S. Schmidt contributed reporting.