That deal came after a string of others that have come under more scrutiny as Mr. Giuliani has pursued his work on Mr. Trump’s behalf. He was paid $500,000 late last year by a company co-owned by a Ukrainian-American businessman who played a key role in facilitating the campaign to pressure Ukraine that Mr. Trump championed.

Mr. Giuliani was paid what he described in an interview as “a reasonable amount of money” in 2017 by a Ukrainian-Russian developer to create an emergency management plan for the developer’s hometown in northeast Ukraine near the Russian border.

And Mr. Giuliani’s security company this year won a contract to consult for the Bahraini government, which described him as leading a “high-level United States delegation” when he visited to pitch his services to the king in December.

While Mr. Giuliani insists he does not lobby, and says his contracts explicitly state that he will not, some of his clients and prospective clients said in interviews that they saw him as a conduit to the administration.

For instance, while Mr. Giuliani was exploring work in the Democratic Republic of Congo last year, its ambassador to the United States said in an interview that his country was relying on Mr. Giuliani to act as a liaison as it sought to avoid further sanctions from the Trump administration.

He has at times sought to shape American foreign policy to benefit his clients, pressing Mr. Trump and Rex Tillerson, then the secretary of state, during an Oval Office meeting in 2017 to consider releasing a jailed client, an Iranian-Turkish gold trader, as part of a potential prisoner swap with Turkey.

The scrutiny of Mr. Giuliani by federal prosecutors goes to whether some of his activities broke foreign lobbying laws. The prosecutors are looking in particular at Mr. Giuliani’s efforts to undermine the American ambassador to Ukraine then, Marie L. Yovanovitch, one person familiar with the case has said. The question would be whether he did so at the behest of, or to benefit, Ukrainian officials with whom he worked, some of whom had been intensely critical of Ms. Yovanovitch. Mr. Giuliani has denied any wrongdoing.