President Trump's push to include a payroll tax cut in the next coronavirus bill is being greeted with skepticism by members of the Senate GOP leadership. 

Trump and members of his administration, including Treasury Secretary Steven Mnuchin, are publicly pushing to include the tax cut, with the president warning there will not be another coronavirus relief bill without it. 

But asked about the payroll tax cut, Sen. John Thune (S.D.), the No. 2 Senate Republican, said "we'll see," adding that a "payroll tax cut only helps if you're on the payroll," in an apparent reference to the tens of millions who have recently filed for unemployment.

"I'm not a particular fan of that. We looked at it before. ... I guess I'm open to being persuaded that it is something that could be effective, but I think some of the things that we're currently doing are having a bigger impact," Thune said. 

Sen. Roy Blunt (R-Mo.), another member of GOP leadership, added that "I'm not persuaded that that's the best way to move forward." 

"What we need to do now is look at what we've done," Blunt said. "I think we ought to evaluate whether we're still where we were six and eight weeks ago."

Sen. John Cornyn (R-Texas), an adviser to Senate Majority Leader Mitch McConnell (R-Ky.), called Republicans “divided” on the payroll tax cut.

Trump has repeatedly floated the idea of a payroll tax cut, but so far lawmakers have refrained from including it in their coronavirus relief packages. 

Trump appeared to make the inclusion of a payroll tax cut a red line in the upcoming negotiations on the next stimulus bill, which would be the fifth piece of legislation passed by Congress in response to the coronavirus.

“We’re not doing anything unless we get a payroll tax cut. That is so important to the success of our country," Trump said during a virtual town hall with Fox News on Sunday. 

McConnell has yet to say if he supports including the payroll tax cut in upcoming legislation.

The GOP leader has drawn his own red lines for the upcoming bill, saying it has to include expanded liability protections for employers who are saying they could be sued as states start to try to lift social distancing restrictions. 

In addition to a lukewarm reaction from key senators, House Speaker Nancy Pelosi (D-Calif.) has called a payroll tax cut a nonstarter.

“Nobody’s putting anything on the table and saying, ‘Unless we have this, we’re not doing that.’ He shouldn’t either,” she told CNN on Monday.