At the same time, the Senate panel is weighing whether to compel the appearance of Trump's former campaign chairman, Paul Manafort, along with two senior FBI officials, all of whom have yet to agree to be interviewed despite the demands of committee leaders.
The series of moves are the latest sign in the uptick of activity by the powerful committee, which is investigating political interference with the FBI, the firing of former FBI Director James Comey
as well as any Russian-Trump campaign coordination in the 2016 elections -- something the President has dismissed as a hoax.
Now the panel plans to begin examining two bills that could prevent Trump from firing the special counsel, something the White House says the President isn't considering, but what many on the Hill fear could still ultimately happen given Trump's lingering frustration with the investigation.
Sen. Thom Tillis, a Republican from North Carolina who is co-authoring one of the bills, said he's been informed that the Senate judiciary committee plans to hold a hearing on the measures within the next two weeks. Two other sources with direct knowledge of the discussions confirmed the account of Tillis, who also sits on the judiciary committee, with the hearing expected in the last week of September.
That is the clearest sign yet that Senate Judiciary Chairman Chuck Grassley has not ruled out calling up the bill for a vote.
"I think he's open," Tillis told CNN of Grassley. "The chair is doing a good job. .... He's giving us a chance to talk about it."
Tillis, who said he had a "pleasant conversation" with Trump about the bill after he introduced it earlier this summer, says he's pushing the measure to "remove distractions."
"Congress has to reassert some checks and balances," Tillis said. "I'm doing this purely from an institutional perspective."
A Grassley spokesman would not comment.
At issue are two bipartisan bills aimed at protecting the special counsel: One from Tillis and Democratic Sen. Chris Coons of Delaware and the other from Sens. Lindsey Graham, a South Carolina Republican, and Cory Booker, a New Jersey Democrat.
The bill from Tillis and Coons would only allow the attorney general -- or the most senior non-recused Justice official -- to remove the special counsel. It would also give the special counsel the chance to challenge the firing in court before a panel of three federal judges.
The legislation from Graham and Booker, on the other hand, would prevent the special counsel from being fired unless the attorney general goes first before a three-judge panel and the court finds there was "misconduct, dereliction of duty, incapacity, conflict of interest or other good cause."
Coons told CNN that he's "very optimistic" of his legislation's chances, given the committee's top Democrat Dianne Feinstein's "strong statements" and Grassley's "balanced and reasonable approach to this investigation."
Indeed, Grassley and Feinstein are moving forward on their own investigation after Donald Trump Jr., the President's eldest son, met with Senate judiciary staff in a closed session last week
. It's unclear if there will be a public session, and Grassley has indicated he wants to interview Manafort before any public hearing with Trump Jr. Both Kushner and the President's son attended a June 2016 meeting
at Trump Tower with Russian operatives, who had promised Trump Jr. dirt on the Clinton campaign.
But Grassley expressed frustration with Manafort.
"I'm not very satisfied with the fact that their lawyers aren't returning our calls," Grassley said, who wanted to discuss the matter with Feinstein before moving forward with any subpoenas compelling Manafort's appearance.
A Manafort spokesman declined to comment. Manafort has provided documents to the committee and interviewed with the Senate intelligence committee in late July.
At the same time, Grassley and Feinstein are trying to push the Justice Department to allow the committee to interview two senior FBI officials, James Rybicki and Carl Ghattas, both of whom could provide first-hand accounts about the firing of Comey. The Justice Department has blocked that request, citing Mueller's investigation.
Asked Thursday if he'd subpoena the FBI officials to compel their appearance, Grassley said: "We're in the process of two steps on that. One is in regard to the counsel, Senate counsel giving us the right language to use, and then I've got to talk to Sen. Feinstein about it."