Republicans "may well" back a pared-down jobs bill set for a Senate vote this week, their leader said Sunday.

Senate Minority Leader Mitch McConnell (R-Ky.) said that GOP lawmakers could end up supporting a $15 billion package set for a vote, even after expressing frustration toward Senate Majority Leader Harry Reid (D-Nev.) pulled an earlier, larger version of the bill that was negotiated on a bipartisan basis.

"We may well," McConnell said during an appearance on "Fox News Sunday" when asked if GOP senators would back the package.

The bill is set for a vote on Monday, and Democrats, having lost their 60-vote majority with the special election of Sen. Scott Brown (D-Mass.), and having Sen. Frank Lautenberg (D-N.J.) absent to begin cancer treatment, will need to rely on Republicans to help pass the package.

Reid pulled the larger bill, which had been negotiated by Senate Finance Committee Chairman Max Baucus (D-Mont.) and Ranking Member Chuck Grassley (R-Iowa), out of concern that Republicans would balk at supporting the plan, or seek to slow down its progress.

"What was a mystery to us is how the bipartisan bill got shelved. I thought it was moving along a bipartisan path," McConnell responded Sunday. "Many of my members were going to support it. And all of a sudden the majority leader decided to skinny it down."

The minority leader denied that the GOP would have sought to slow down the jobs bill's progress, saying they had intended to move forward with the original package as quickly as lawmakers plan to move ahead with the current proposal.

Reid said this weekend that at least one Republican, Scott Brown, has signaled there's a good chance he'll support the bill.