Grassley and three other Republicans voted in favor of it.
Special Counsel Robert Mueller, head of the probe looking into Russian interference in the 2016 presidential elections.
The bill, sponsored by Tillis and Graham (R-S.C.) with Sens.
Some senators have said the measure could become an insurance policy - and perhaps the starting point for a congressional response - if Trump defies their advice and acts to fire Mueller.
A senior Democrat on the committee, Senator Patrick Leahy, tweeted that Trump's "rant this morning undermines the rule of law", and Trump "already improperly interfered with [the Justice Department] more than any president since [Richard] Nixon".
But Cornyn, the number two Republican in the senate, made a stark prediction "This bill will not be taken up on the Senate floor".
"I don't think it would end the investigation". Still, he warned that "firing Mueller would cause a firestorm and bring the administration's agenda to a halt, and could even result in impeachment (of Trump)".
Four Republicans - Grassley, Graham, Tillis and Sen. Should the court find that the firing was not for good cause, the special counsel must return to their job.
Victor Oladipo: LeBron James' Clutch Block a Goaltend
Wednesday's shot was reminiscent of the game-winner he hit in Game 2 of the conference finals against Orlando in 2009. Apart from a short blip on the radar in Games 1 and 3, Korver has been putting up 3 after 3 for the Cavaliers.
Two Republicans and two Democrats introduced the bill earlier this month as Trump ramped up criticism of the special counsel.
"I'm very disappointed in my Justice Department", he said.
The measure may die in committee, however, as the full Senate may not vote on it, according to Senate Majority Leader Mitch McConnell.
"It's possible the bill goes too far", he said.
Sasse told the Standard that although he doesn't support the current bill, he does support a non-legislative resolution that Mueller "should be permitted to finish his work in a timely fashion".
Almost every Republican on the panel said they did not think Trump should take steps to fire Mueller.
But the attorney general's comments, nonetheless, are striking, suggesting Sessions might privately share some of the president's frustration with the Mueller investigation. They urged the president to allow the investigation to continue. "I think that's in the best interests of the president and the country", said Orrin Hatch, R-Utah. "We took an oath to protect and defend the Constitution of the United States".
Removing Mueller or Rosenstein with the goal of removing Mueller "would blow up in his face". That decision has not been overturned, yet it includes a renowned dissent from Justice Antonin Scalia that says Congress can't infringe on a purely executive power to prosecute.
A newly formed Republican group aired ads in North Carolina, South Carolina, Iowa and Kentucky this week encouraging Republicans to support the legislation.