Senate Republicans were initially skeptical of the idea of a “rescissions” package to roll back federal spending, with some expressing concern about going back on the $1.3 trillion spending deal they had just reached with Democrats.

“You can’t make an agreement one month and say: 'OK, we really didn’t mean it,’” Senate Majority Leader Mitch McConnell told Fox News last month.

But McConnell and other Senate Republicans sounded a lot more open to the $15.4 billion rescissions package the White House unveiled Tuesday, which does not seek to make cuts from March’s omnibus spending package.

"My understanding of the rescission package is that it does not breach the bipartisan agreement we reached in the caps deal. If the House is able to pass the rescissions package, we'll take a look at it," McConnell said Tuesday, according to The Hill.

But with a slim 51-49 majority in the Senate, and with Sen. John McCain away from Washington because of his cancer battle, Senate Republicans can’t afford to lose any votes, unless they manage to get some Democratic support for the spending cuts.

A number of Republican senators have said they still need to look at the details of the proposed cuts before deciding how they’ll vote, though Sens. Lisa Murkowski (R-AK) and Shelley Moore Capito (R-WV), among others, have already voiced some concerns about the proposed cuts to the Children’s Health Insurance Program.