The F.B.I. request could be a routine, “check-the-box” request, according to one person briefed on the request. But it also could be related to the more substantive issues about the Ukraine pressure campaign raised in the whistle-blower complaint, according to people familiar with the matter.

Yahoo News first reported that the F.B.I. was seeking to speak with the whistle-blower.

In August, after learning about the whistle-blower complaint, the Justice Department’s criminal division reviewed the whistle-blower complaint and found that there was no violation of campaign finance laws. The next month, as the complaint became public, the Justice Department announced that it had determined no further action was necessary on the complaint.

Mr. Giuliani is under scrutiny by federal prosecutors in Manhattan, at least in part for his role in efforts to remove the United States ambassador to Ukraine, Marie L. Yovanovitch. Mr. Giuliani was also a key player in the efforts to press Ukraine’s government to pursue the investigations sought by Mr. Trump in his July 25 call with President Volodymyr Zelensky of Ukraine. They included investigations into former Vice President Joseph R. Biden Jr. and a debunked theory about Ukrainians meddling in the 2016 election.

The F.B.I. might also have wanted to learn about threats made against the whistle-blower. The C.I.A. is providing protection to the whistle-blower, who is an agency employee continuing to work on intelligence matters.

Still, the F.B.I. has not renewed its request to speak to the whistle-blower since it first asked several weeks ago, according to a person familiar with the matter.