The Senate Judiciary Committee appears poised to vote in two weeks on a bill that would give job protections to Special Counsel Robert S. Mueller III, even as President Donald Trump asserted again Thursday that he has the authority to fire the man investigating connections between Trump’s campaign and Russian operatives.

Thursday’s discussion revealed how the bill still faces potential political hazards at the Judiciary Committee. Democrats have raised concerns about a yet-to-be-seen amendment that Republicans want to add to the measure. Some Republicans have concerns about the constitutionality of a bill that would limit a president’s ability to make personnel decisions in the executive branch.

The bill faces additional hurdles even if it overcomes opposition on the Judiciary Committee. Senate Majority Leader Mitch McConnell, who controls what bills reach the floor, has said such a bill is not needed. The Republican-led House might not support such a bill, and even if it did, Trump himself could veto it.

Chairman Charles E. Grassley, R-Iowa, said he still has concerns about the constitutionality of the bill, but has a responsibility to put it on the agenda because it is a bipartisan compromise among four committee members: Republicans Lindsey Graham of South Carolina and Thom Tillis of North Carolina and Democrats Chris Coons of Delaware and Cory Booker of New Jersey.

The measure combined two bills, first introduced eight months ago, and give a special counsel 10 days to ask a federal judge to review whether a removal was for “good cause,” and if not, allow the special counsel to stay in the job.

“It’s clear that the constitutionality of these bills is something the court would have to resolve,” Grassley said.

Sen. Dianne Feinstein of California, the panel’s top Democrat, said there is bipartisan agreement that Mueller needs to be able to finish his work without political interference, and that she supports the bill and wants it to “be considered as soon as possible.”

Yet Feinstein did not agree to put the bill on Thursday’s agenda, which would have allowed a vote next week, because of an amendment Democrats haven’t been able to review that they say could undermine the investigation.

“There should be no political pressure brought to bear on the special counsel by the White House or by Congress,” Feinstein said.

Grassley called those concerns “unfounded.” The Iowa Republican said he plans to offer an amendment that would “require the attorney general to give a detailed report to Congress justifying significant decisions involving the special counsel, including the firing of the special counsel.”

Potential Amendment

Tillis, one of the bill’s authors, said some of the possible amendment “is increased transparency to address some of the concerns that we have with some of the actions of some people in the Department of Justice that didn’t seem appropriate.”

After the hearing, Sen. Richard Blumenthal, D-Conn., told reporters he has concerns with any amendment that would weaken the protections of the special counsel in the bill. He said he wants to closely scrutinize the language about the report to make sure it is not a substitute for the protections in the special counsel bill.

“If they are simply surplusage, if they are in addition to enforceability, that’s fine,” Blumenthal said. “But there should be no language that in effect compromises a court order that would protect the special counsel.”

Grassley, in a written statement put in the record, said his amendment would not change the special counsel bill. “I’m at a loss to see how a call for the administration to be more transparent about decisions involving the special counsel — including any decision to fire the special counsel or curtail his investigation — would undermine the Mueller investigation,” Grassley said.

Sen. Richard J. Durbin, D-Ill., raised the possibility that, since the vote on the bill may not be for two weeks, there should be a “sense of the committee resolution” to make clear the committee’s position on the termination of Mueller and Deputy Attorney General Rod Rosenstein. Some legal scholars say Trump could choose to fire Rosenstein because he is the Justice Department official who oversees the special counsel probe.

Grassley quickly shot down Durbin’s request because he didn’t tell Republicans about it ahead of time.

About four hours before the committee met, Trump tweeted about the investigation again: “If I wanted to fire Robert Mueller in December, as reported by the Failing New York Times, I would have fired him. Just more Fake News from a biased newspaper!”

And after the committee meeting, Trump again took to Twitter: “I have agreed with the historically cooperative, disciplined approach that we have engaged in with Robert Mueller (Unlike the Clintons!). I have full confidence in Ty Cobb, my Special Counsel, and have been fully advised throughout each phase of this process.”