The United States attorney’s office in Manhattan and the F.B.I. had sought for months to secure a search warrant for Mr. Giuliani’s phones. Under Mr. Trump, senior political appointees in the Justice Department repeatedly sought to block such a warrant, The New York Times reported, slowing the investigation as it was gaining momentum last year. After Merrick B. Garland was confirmed as Mr. Biden’s attorney general, the Justice Department lifted its objection to the search.While the warrants are not an explicit accusation of wrongdoing against Mr. Giuliani, it shows that the investigation has entered an aggressive new phase. To obtain a search warrant, investigators need to persuade a judge they have sufficient reason to believe that a crime was committed and that the search would turn up evidence of the crime.Spokesmen for the F.B.I. and the U.S. attorney’s office declined to comment.The investigation of Mr. Giuliani grew out of a case against two Soviet-born men who aided his mission in Ukraine to unearth damaging information about Mr. Biden and his son Hunter, who was on the board of a Ukrainian energy company. The prosecutors charged the two men, Lev Parnas and Igor Fruman, with unrelated crimes in late 2019 and a trial is scheduled for October.While investigating Mr. Giuliani, prosecutors have examined, among other things, his potential business dealings in Ukraine and his role in pushing the Trump administration to oust the American ambassador to Ukraine, which was the subject of testimony at Mr. Trump’s first impeachment trial.As he was pressuring Ukrainian officials to investigate the Bidens, Mr. Giuliani became fixated on removing the ambassador, Marie L. Yovanovitch, whom he saw as an obstacle to those efforts. At the urging of Mr. Giuliani and other Republicans, Mr. Trump ultimately ousted Ms. Yovanovitch.As part of the investigation into Mr. Giuliani, the prosecutors have explored whether he was working not only for Mr. Trump, but also for Ukrainian officials or businesses who wanted the ambassador to be dismissed for their own reasons, according to people briefed on the matter.