Senate Majority Leader Mitch McConnell said Republicans are eyeing a new targeted coronavirus relief bill based on what President Trump would sign into law, and poured cold water on a new bipartisan $908 billion proposal his colleagues put forth earlier Tuesday.

"We just don't have time to waste time," McConnell, R-Ky., said of the compromise legislation a group of Democrats and Republicans released Tuesday. "We have a couple of weeks left here. Obviously, it does require bipartisan support to get out of Congress, but it requires a presidential signature. And this government is in place for sure for the next month. And I think the place to start is: Are we actually making a law or are we just making a point?"

McConnell said he talked with Treasury Secretary Steven Mnuchin to ascertain exactly what Trump would sign into law. That proposal -- pegged at roughly $500 billion -- was circulated to Senate Republicans Tuesday and will be the basis of the next GOP COVID relief effort. 

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A copy of the proposal obtained by Fox News shows the legislation includes liability protections for businesses, more than $300 billion in new Paycheck Protection Program money for employers, $105 billion for schools to reopen and an extension of unemployment benefits through Jan. 31 with a two-month phase-out. Not included in the legislation is another round of direct stimulus checks of $1,200, rental assistance or state and local funding. 

The timeline for the coronavirus relief bill is around the corner. Government funding ends on Dec. 11 at 11:59 p.m., and Congress must pass a new spending bill to avert another shutdown. McConnell said the $1.4 trillion omnibus spending bill could be the "vehicle" to add on the latest coronavirus relief legislation.

"I think it all might become one package," McConnell said.  

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Meanwhile, Sen. Chuck Schumer, D-N.Y., blasted McConnell for the "partisan" approach and urged the majority leader to work with Democrats. 

Schumer and House Speaker Nancy Pelosi, D-Calif., sent McConnell a new offer Monday night to get the ball moving on bipartisan talks. The details of their offer were not made public.

"The biggest impediment to getting an agreement is the Republican leader refusing to negotiate in a bipartisan way," Schumer said. "He knows darn well the House is a Democratic majority. He knows darn well he needs Democratic votes in the Senate ... and yet he continues to negotiate in a partisan way."

Earlier Tuesday the House Problem Solvers Caucus and a bipartisan group of senators released their joint plan for coronavirus relief in an effort to bridge the gap and find common ground.

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"It won't make everyone happy," Sen. Mark Warner, D-Va., said during a press conference on Tuesday. "But there's been an enormous amount of work done."

Their $908 billion bipartisan agreement allocates about $300 billion in funding for small businesses through the Paycheck Protection Program, $240 billion in aid for state and local governments, $180 billion to extend boosted unemployment benefits at $300 per week for four months and a temporary moratorium on COVID liability lawsuits to allow states enough time to design their own laws. It notably does not include a second stimulus check.

Mike Emanuel and Megan Henney contributed to this report.