Washington (CNN) Senate Judiciary Chairman Chuck Grassley's amendment to legislation protecting the special counsel does not include real-time reporting requirements about changes to the scope of the investigation, which was a chief complaint from Democrats about his initial proposal.

Grassley's 20-page amendment was posted on Wednesday evening ahead of a his committee's planned markup Thursday of bipartisan legislation that would provide Robert Mueller and future special counsels a path to challenge their firing in court.

The tweaked amendment from Grassley came as other Republicans who are opposed to the legislation prepared their own plans for the markup. The second-ranking Senate Republican, Sen. John Cornyn of Texas, introduced an amendment Wednesday to have a "sense of the Senate resolution" warning the White House not to fire Mueller, instead of a bill to formally protect the special counsel.

"I think that might be a way forward because it avoids the unconstitutionality issue on a bill the President won's sign and the House won't pass," Cornyn told reporters. "So, that may be a place for us to land. Because, as I've said, I think it would be a mistake for the President to fire Director Mueller and I think if we can come to some common ground on a resolution, that might be the path forward."

Cornyn's amendment states that Congress "should not resurrect unconstitutional barriers to executive authority and weaken the separation of powers in the name of political expediency," and also that "Robert Mueller should be permitted to finish his work in a timely fashion."

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