Mr. Giuliani would not comment Monday on whether he had shared any information with the Justice Department, but he rejected questions about the credibility of what he had collected. “My information checks out 10 ways to Sunday,” he wrote in a text message, asserting that he had obtained four or five “unquestionably true documents” relating to the Bidens’ work in Ukraine.

“More is getting discovered as we speak,” he added.

Federal prosecutors outside of Washington are vetting the materials from Mr. Giuliani, a person familiar with the matter said.

This is not the first time that law enforcement officials under the Trump administration have farmed out politically toxic information far from the main Justice Department blocks from the White House. Former Attorney General Jeff Sessions asked John W. Huber, the United States attorney in Utah, to examine allegations by Mr. Trump and his allies about Hillary Clinton. Though Mr. Huber is not known to have closed that years-old inquiry, he has also not filed any charges.

In the past, the Justice Department has distanced itself from Mr. Giuliani. In September, a spokeswoman said that Mr. Barr had “not discussed anything relating to Ukraine” with Mr. Giuliani.

And a spokesman has said that Brian A. Benczkowski, the assistant attorney general for the department’s criminal division, would not have included Mr. Giuliani in an August meeting about a bribery case had he known that federal prosecutors in New York were investigating two of Mr. Giuliani’s associates, Lev Parnas and Igor Fruman.

Mr. Parnas and Mr. Fruman were charged weeks after the meeting with violating campaign finance laws to try to unlawfully influence politicians, including former Representative Pete Sessions, Republican of Texas.

Mr. Giuliani had worked with the two men last year to collect information about — and advance investigations into — targets of Mr. Trump, including Mr. Biden, who is seeking the Democratic presidential nomination. Mr. Biden was the Obama administration’s point man in pushing anti-corruption initiatives in Ukraine.