The meeting came as another panel, the Senate Judiciary Committee, announced that it issued a subpoena for Mr. Manafort to appear at a hearing on Wednesday. But the committee later rescinded the subpoena and canceled his appearance. Mr. Manafort’s lawyers are now working out how and when he will be interviewed by the committee. The panel is conducting its own investigation into possible ties between Mr. Trump’s campaign and the Russian government.

Mr. Manafort has been at the center of inquiries into whether Mr. Trump’s senior advisers coordinated with Kremlin efforts to disrupt last year’s election.

Mr. Manafort met with investigators a day after the president’s son-in-law and senior adviser, Jared Kushner, provided his own account of the June 2016 meeting to them. The meeting at Trump Tower was set up by Donald Trump Jr., the president’s eldest son, who had been told by an intermediary that the Russian lawyer, Natalia Veselnitskaya, had damaging information about Mrs. Clinton. Mr. Kushner met with the House Intelligence Committee, which is also conducting an investigation, on Tuesday.

The subpoena announcement was a sign of a brewing turf battle between congressional committees conducting their own investigations into the election. The Senate Intelligence Committee, which began investigating the issue at the beginning of the year, had offered to provide a transcript of Mr. Manafort’s Tuesday interview to the Judiciary Committee.

The Judiciary Committee’s top two senators rejected the offer.

“Mr. Manafort, through his attorney, said that he would be willing to provide only a single transcribed interview to Congress, which would not be available to the Judiciary Committee members or staff,” a committee news release said on Tuesday. “While the Judiciary Committee was willing to cooperate on equal terms with any other committee to accommodate Mr. Manafort’s request, ultimately that was not possible.”