“Republicans need to look in the mirror and do a gut check,” she said.

At the moment, Republican leaders seem to be in no mood to add to the furor surrounding the White House by forging ahead with such a special panel, pointing instead to what Senator Mitch McConnell, Republican of Kentucky and the majority leader, called the “broad discretion” the committee has to look at intelligence matters.

Creating a special committee or commission would require the consent of both parties, and the idea is more likely to gain traction if the Intelligence Committee or others in Congress pursuing questions about Russia uncover compelling evidence that would make it hard to resist demands for a deeper, more public investigation. The special national commission on the Sept. 11, 2001, attacks was created only after a special joint investigation by the intelligence committees of the House and Senate conducted its own inquiry that raised new questions.

Members of the intelligence panel have multiple motivations for taking a forceful stance after Mr. Flynn’s resignation. Given heightened attention to the issue, lawmakers know they risk losing control of the investigation if they are seen as not being aggressive. Neither Democrats nor Republicans who sit on the committee — their work often unsung with constituents back home because of secrecy — want to see that.

Senator John McCain, Republican of Arizona, who leads the Armed Services Committee, is also examining the administration’s relationship with Russia, though the intelligence committee has more official responsibility.

On Tuesday, Mr. McConnell said it was “highly likely” the committee’s investigation would expand beyond election interference to include Mr. Flynn’s activities. Senator Roy Blunt, Republican of Missouri, who sits on the Intelligence Committee, predicted Mr. Flynn would be called before the panel to testify “about both postelection activities and any other activities that he would be aware of.”

Mr. Blunt, a member of the leadership with sharp political instincts, will be a Republican committee member to watch along with fellow Republican senators like Susan Collins of Maine, a centrist who has challenged Mr. Trump on other issues, and Marco Rubio of Florida, a sometime Trump critic who has urged the administration to take a hard line against Russia.