Washington (CNN) Senate Judiciary Chairman Chuck Grassley urged President Donald Trump Friday to let Robert Mueller's investigation "work its course," adding that he was "open" to considering bipartisan legislation aimed at protecting the special counsel.

In an interview with CNN, Grassley said he did not believe Trump would actually fire Mueller, despite reports Thursday night that the President took steps last June to dismiss the special counsel and later backed off after White House Counsel Don McGahn threatened to quit.

Grassley said the reports show Trump listens to his staff.

"I just don't think the President -- as unpredictable as he is -- would fire Mueller, and I take the view, and I said so maybe not directly to the President, but indirectly to the President: Just let this work its course," Grassley said.

Asked if he would be OK if Trump fired Mueller, Grassley said: "Heavens no."

The comments reflect the wide support Mueller has on Capitol Hill, with many Republicans warning privately and publicly there would be a backlash if Trump took that dramatic step.

Bills to protect special counsel

Democrats are trying to renew efforts to push bipartisan legislation in the Senate to give Mueller and future special counsels protection from political pressure from the White House. There are two competing bills at the moment.

"I'm surely open to considering those bills," Grassley said, adding that the two approaches should be reconciled and then he would focus on whether they raise any constitutional concerns about the separation of powers between the branches of government.

The measures face an uphill battle through Congress, where Senate Majority Leader Mitch McConnell and other Republicans have downplayed the need for such legislation. But Democrats -- led by Sens. Cory Booker and Chris Coons -- are trying to renew the efforts in light of the new reports.

"It is more important than ever for Congress to act to protect the independence of the Department of Justice, including the special counsel investigation," said Coons of Delaware, who partnered with Sen. Thom Tillis, R-North Carolina, on one of the special counsel bills. "These reports make clear that we need to act."

Booker told CNN Friday reports that Trump was trying to fire Mueller show his "authoritarian tendencies" and says there is now a "moral imperative" to pass bills to shield the special counsel from political pressure. He said he's renewing talks with Republicans to move the bills.

"I just hope that Congress now rallies around the idea that here is a good piece of legislation to make sure that we have a check-and-balance in the judiciary branch, on the President's powers," Booker said. "I would say that reactions before this broke were, hey, this is a good bill. This is good legislation. But what's the hurry? What's the urgency? Now, I think what we're going to hear is a more sober analysis of how to move forward."

After Tillis announced his bill last summer, Trump called him on the phone and asked him about his tactics, which Tillis defended, the lawmaker told CNN. But in recent weeks, as talks about firing the special counsel had subsided, the North Carolina senator has not focused on the measure as his attention has shifted to immigration and other matters.

Tillis spokesman Daniel Keylin said some members have raised constitutionality concerns about the bills, and they currently do not have the support needed to move them through Congress. Keylin said Friday that the talk about removing Mueller since last August when the measures were introduced have "completely come to a halt" as the President's team has shown signs of cooperation with the special counsel.

The Tillis and Coons bill would give a special counsel the chance to challenge the firing in court before a panel of three federal judges. It also would only allow an attorney general who has been confirmed by the Senate — or senior non-recused, Senate-confirmed DOJ official— to fire the special counsel.

A bill from Booker and South Carolina Republican Sen. Lindsey Graham instead would require the attorney general to first get approval from a three-judge panel before the special counsel could be fired. In their measure, a special counsel would only be fired "after the court has issued an order finding misconduct, dereliction of duty, incapacity, conflict of interest, or other good cause."

McConnell was asked about the Mueller protection bills in the fall, and he said that the Senate was working on other priorities through the end of the year, adding that the chamber should be focused on its own Russia meddling investigation.

"We're going to concentrate on what we're doing here in the Senate," McConnell said.