Committee and staff members have been hesitant to put a timetable on the investigation, saying that additional document and witness requests are likely in the coming weeks. Just how far Mr. Grassley, who as chairman has considerable power over any investigation by the committee, is willing to go remains a matter of speculation. Democrats said they were cautiously optimistic that they would be able to advance in a bipartisan fashion.

“I think the chairman has been a very straight shooter,” said Senator Richard Blumenthal, Democrat of Connecticut and a member of the panel. “He has a very serious commitment to uncovering the truth.”

Committee members originally wanted Donald Trump Jr. to testify publicly before the committee, along with President Trump’s former campaign chairman, Paul Manafort, and Glenn Simpson, the founder of the firm that produced a dossier of sensational claims about President Trump and Russia. But Donald Trump Jr.’s lawyers negotiated an agreement for the closed-door meeting and the production of communications related to the meeting.

The interview on Thursday was expected to be largely handled by staff, though several senators said they planned to attend. Besides the Trump Tower meeting, Mr. Blumenthal said the staff would question Mr. Trump on the Trump Organization’s financial dealings with the Russians and on his father’s decision to fire Mr. Comey.

Mr. Trump set up the June 2016 meeting after he had been told by an intermediary that a Russian lawyer, Natalia Veselnitskaya, had damaging information about Mrs. Clinton to share. Also in attendance were Mr. Manafort and Jared Kushner, the president’s son-in-law, now a White House adviser.

Mr. Manafort and Mr. Kushner have both met with investigators from the Senate Intelligence Committee. Those interviews have been the cause of some tension between the two Senate panels, and Judiciary members said they were still hopeful that they, too, would get a chance to question the men.

At the same time, the Judiciary Committee has been in consistent communication with Robert S. Mueller III, the special counsel investigating Mr. Trump’s Russia ties and related matters, to make sure the two investigations did not overlap or cause legal issues down the road, committee aides said. The special counsel’s team has also made copies of Mr. Comey’s written memos available to Mr. Grassley and Ms. Feinstein, who have continued to press for their own copy.