The letter to Mr. Stone was signed by the committee’s chairman, Senator Richard M. Burr, Republican of North Carolina, and its ranking Democrat, Senator Mark Warner of Virginia. Press officers for Mr. Burr and Mr. Warner declined to comment on the letter.

Democrats and some investigators, as well as some Republicans, have been watching Mr. Stone, a Richard M. Nixon acolyte and self-described “dirty trickster,” more closely since he posted on Twitter in August 2016 about John D. Podesta, Hillary Clinton’s campaign chairman, whose private emails were hacked and provided to WikiLeaks.

Mr. Stone said on Twitter that Mr. Podesta would soon face his “time in the barrel,” two months before the emails were made public. Mr. Stone maintained that he was alluding to business activities he attributed to Mr. Podesta, not prior knowledge of the hackings.

Mr. Stone said he is eager to provide the committee with information.

“I am anxious to rebut allegations that I had any improper or nefarious contact with any agent of the Russian state based on facts, not misleading and salacious headlines,” he said, adding, “I am willing to appear voluntarily if the committee isn’t looking for the headline of issuing a subpoena.”

Mr. Stone, who has strenuously denied the allegations for months, has retained two lawyers to assist with his response to the inquiry, as well as in the hope of pushing federal investigators to either make their information public or say that no case exists.