McConnell hopeful Senate can finish Violence Against Women reauthorization on Thursday

"This approach is in keeping with how Republicans have handled VAWA in the past," he said. "This approach would also allow us to complete the bill today."

Senate Majority Leader Harry ReidHarry ReidSenate Dems accuse GOP of slow-walking Obama nominees The Trail 2016: GOP stages of grief Dems slam Trump over taco bowl tweet MORE (D-Nev.) spoke just before McConnell, and said the two leaders were still trying to work out how to proceed with amendments. Reid said he hoped senators could announce an agreement by about 2 p.m.

McConnell said Republicans are interested in strengthening the law by making it easier to put people who abuse women in jail.

"As my friend the majority leader noted yesterday, a good way to lower the incidence of violent crime is incarcerate those who commit it," he said. "We couldn't agree more, and we'd like the chance to improve the law in that respect."

Senate Democrats have been pressing to complete work on a bill reauthorizing the Violence Against Women Act (VAWA), to the point that Republicans have accused Democrats of politicizing an issue even though there is no disagreement about reauthorization.

Some differences have emerged, however, over how to reauthorize the program. For example, Republicans are known to oppose the Democratic bill in part because it would increase the deficit.

An alternative GOP bill from Sen. Kay Bailey Hutchison (R-Texas) would maintain the current number of visas available to aliens who need non-immigrant status to access the help of law enforcement authorities in domestic abuse cases. Republicans said the Democratic proposal to expand these visas would lead to the $105 million in new deficits under the bill.

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