The bipartisan legislation was introduced last week as Trump publicly criticized Mueller, who is investigating potential ties between Russian Federation and Trump's 2016 campaign as well as possible obstruction of justice by the president.
The measure is on the Judiciary Committee's agenda for Thursday, but due to the panel's rules the bill will be marked up the following week.
McConnell's determination that the action is not needed is apparently regardless of what happens in the Senate Judiciary Committee.
Trump, however, has pledged in the past not to dismiss Mueller. McConnell hasn't said boo about Trump's foreign emoluments, his grotesque conflicts of interest, or the nepotism and self-enrichment that are endemic to his administration.
"While I'm glad the majority leader believes the President would be wrong to fire Special Counsel Mueller, it's a mistake not to pass legislation to protect the investigation", he said in a statement.
Tillis suggested that McConnell could be eventually convinced to change course. As one of my astute Twitter followers pointed out, the majority leader's stance is akin to refusing to buy vehicle insurance because you have no plans to get into an auto accident anytime in the future. "I don't think Mueller is the reason for passing this bill".
McConnell's reaction was quite different - in the Fox interview, he questioned why Congress would expend effort on trying to get a bill passed that the president was unlikely to sign.
"Just as a practical matter, even if we pass it, why would he sign it?" McConnell said, adding: "This is a piece of legislation that's not necessary, in my judgment".
Grassley said no senator has talked to him directly to complain about considering the bill.
"I think this bill has as much to do with decades into the future as it does just with Mueller, because I don't think Mueller's going to be fired", said Grassley of Iowa. Dianne Feinstein, the top Democrat on the panel, ahead of Thursday's meeting, and he planned to provide it to the full committee membership later Thursday.
"As I stated previously, I'm concerned about requiring law enforcement officials to report prosecutorial decisions during open criminal investigations", California Sen.
Leaving the hearing, Feinstein said she was six pages through the 21-page amendment, and wanted to finish reading it before commenting further.
"I think it's probably unconstitutional and I don't think there's any realistic chance that the president will fire Mr. Mueller", Sen.