But legislators are seeking to check executive power by making any indefensible firing of the special counsel far tougher to do.
One bill would provide for a judicial review
The bill -- which is retroactive to May 17, the day Mueller was appointed -- focuses on judicial recourse in case of an unwarranted removal of a special counsel,
The Delaware senator said the legislation would create “a private right of action for a special counsel, empowering him to seek judicial review” in the event of a firing and to get a determination within 14 days of the action being filed.
“This reflects, in my view, a broader bipartisan concern that the president might take inappropriate action to interfere with the ongoing important work of Special Counsel Bob Mueller,” Coons told a small gathering of reporters Thursday.
Sen. Tillis said the bill would provide a check on the president's power to fire a special counsel.
"The president would maintain the power to remove the special counsel, but we would just want to make sure that it had merit and have that back-end judicial process," Tillis told CNN.
Another bill seeks to prevent unwarranted dismissal
Coons said the bill by him and Tillis would work in tandem with the one by Graham and Booker, for he called “a belt-and-suspenders” approach.
The two bipartisan legislative efforts could be merged into one bill in coming weeks but would still have a long way to go before becoming law, including facing any potential presidential veto.
Concern rose among senators of both parties after the president attacked Attorney General Sessions
Republican Sen. Graham warned late last month that any move by Trump to remove Mueller “could be the beginning of the end of the Trump presidency, unless Mueller did something wrong."
Trumps' personal attorney said the president “isn’t thinking about firing Bob Mueller, so this speculation is just incorrect.”
And the new White House chief of staff, John Kelly, has assured Sessions that his job is safe.
But some lawmakers want to try to prevent any potential presidential interference with Mueller’s probe.