The senators behind those bills, Republicans Lindsey Graham of SC and Thom Tillis of North Carolina and Democrats Chris Coons of DE and Cory Booker of New Jersey, combined their efforts for the measure released Wednesday. The original bills were introduced last summer.
He could try to challenge the grounds for his dismissal in court.
Democrat Feinstein said she was anxious about an amendment to the bill that she had not been able to review.
The Senate Judiciary Committee would need to vote on the bill before it could go to the Senate floor for a final vote.
Grassley has not endorsed the bill, and said he has concerns that it is unconstitutional.
Tuesday, the White House declared that President Donald Trump has the power to fire special prosecutor Robert Mueller, stoking fears that the President will put an end to Mr. Mueller in his widening investigation.
Senate Democratic leader Chuck Schumer of NY released a statement warning Trump against firing Mueller.
He is also planning to offer an amendment to the bill requiring the president to give Congress advance notice of his intention and report the reasons for firing a special counsel, much like the requirements for firing an inspector general. The mass protest would also be triggered if Trump moved to replace the deputy attorney general, Rod Rosenstein - which could clear a path for Mueller to be fired - or if Trump pardoned key witnesses in the Russian Federation investigation. And it would also call for a 10-day window of judicial review to determine if a removal is based on "good cause". If a court found that the firing was not for good cause, then the special counsel would stay on the job.
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According to published reports, the release attributed the shift in voters' attitudes to Federal Bureau of Investigation raids on Monday of the home and offices of Trump's personal lawyer Michael Cohen, reportedly due to a tip provided by Mueller.
McConnell's spokesperson declined to comment on Wednesday. "This is not something precipitous, this is something we've been working on".
Since removing a president is a political process, not a legal one, that question would be up to Congress, and the political calculus may change if Democrats gain control of one or both houses in the November midterm elections.
"If there's any realistic opportunity for appropriate legislation, I'm for it", said Sen. Mike Rounds said of the bill. I don't think it's under consideration, because of the kind of conversations I've had with people in the White House, ' he said. Well, this morning, four senators announced a bipartisan bill to protect Mueller.
Sen. Bob Corker, R-Tenn., said it would be "inappropriate" for Trump to dismiss either Mueller or Rosenstein.
Caught off guard and furious with the encroaching inquiry, the president showed a flare of temper watching cable news coverage of the raid Monday afternoon, summoning lawyers Ty Cobb and Jay Sekulow to get their opinion of what was happening.
"The president is not talking about firing Mueller". Mueller, who is heading the DOJ's probe into whether the Trump campaign colluded with the Russians in 2016, forwarded information to local federal authorities that was outside his purview, but might have been felonious in nature.