After he spoke, McConnell took to the floor and said Republicans are happy to speed the process along.

"We'd be very happy to enter into a short time agreement," McConnell said. "We would be happy to work with him to expeditiously approve that bill in short order, and those discussions over some very short time agreement can begin as soon as now."

"I think that's a positive statement, as long as there's not efforts made to weaken this legislation," Reid replied. "If this … short time agreement is an effort to weaken the bill, we want no part of that."

McConnell did not indicate that Republicans would seek dramatic changes to the Democratic bill, S. 1925. "There's no reason to have a fight over something nobody wants to have a fight over," he said.

But Republicans on Monday did introduce an alternative bill, S. 2338, which would address some problems the GOP sees with the current program. For example, the GOP bill would restructure the bill to ensure it does not increase the deficit.

It would also target more money for stopping sexual assault, double the minimum federal sentence for rape to 10 years and create a minimum one-year sentence for possessing child pornography, among other changes.

The GOP bill also limits the number of "U visas" that aliens in the U.S. can use to obtain temporary legal status to get help when they are victims of domestic violence crimes. Democratic bill would allow unused U visas from years past to be accessed.

Republicans cite a Congressional Budget Office estimate that says this language would add $105 million to the deficit, since recipients of U visas could access federal programs like Medicaid. The GOP bill strikes the Democratic proposal and maintains the current annual limit on U visas.

Republicans said their bill includes several changes that enjoy bipartisan support, a sign they might yet look to include some of these provisions in the final bill.