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Soon after calling Mr. Giuliani to say that he might represent Mr. Cohen, Mr. Costello emailed Mr. Cohen and said, “Rudy was thrilled and said this could not be a better situation for the President or you,” noting that Mr. Giuliani knew and trusted Mr. Costello. Mr. Giuliani, he added, “said thank you for opening this back channel of communication.”

Mr. Costello ultimately had a falling out with Mr. Cohen and never formally represented him.

Few details of the current criminal inquiry focused on Mr. Giuliani have been revealed publicly, and he has not been accused of wrongdoing. But the investigation could cover a broad range of his conduct, including his work with two associates and former Ukrainian prosecutors.

Together, the men tried to dig up dirt about former Vice President Joseph R. Biden Jr., a candidate in the 2020 presidential election, and his son Hunter, and increase pressure on other targets of Mr. Trump and his allies, including the American ambassador to Ukraine.

The investigation into Mr. Giuliani appears to have grown out of a 14-month inquiry that has already resulted in charges against the associates and two other men. The associates, Lev Parnas and Igor Fruman, were indicted last month on campaign finance violations, some of which were linked to work the two men did for Mr. Giuliani in Ukraine. They have pleaded not guilty.

One of the federal prosecutors handling that case said in court two weeks ago that the investigation was ongoing, though “no decision about any type of additional charges” had yet been made.

Mr. Giuliani was briefly represented by Jon A. Sale, a former Watergate prosecutor who also worked as a federal prosecutor in the Manhattan office, in connection with a subpoena he received last month from House impeachment investigators.

On Mr. Giuliani’s behalf, Mr. Sale declined to cooperate with the impeachment inquiry, writing on Oct. 15 that the material sought was “beyond the scope of legitimate inquiry” and a violation of attorney-client and executive privilege. On the same day, Mr. Giuliani said on Twitter that Mr. Sale would no longer be representing him because the matter related to the subpoena was over and, “at this time, I do not need a lawyer.”

Ben Protess contributed reporting.