The extent of any assistance that Mr. Giuliani might provide remains unclear and the negotiations could easily fall apart, especially as Mr. Trump continues to publicly rail against the investigation.

But Mr. Giuliani, through his lawyer, has signaled to the committee that he plans to take a less confrontational stance toward its requests than some other members of Mr. Trump’s inner circle who are fighting the committee’s subpoenas or have otherwise refused to cooperate.

Mr. Giuliani’s discussions with committee officials suggest that he may be seeking to avoid a potentially costly legal fight over a subpoena that was issued to him last month. By engaging with the committee, Mr. Giuliani could also make it more difficult for the House to issue a criminal referral of him to the Justice Department for contempt of Congress if he in the end does not comply with the subpoena.