Some Republicans had been critical of the legislation because of several provisions, including one to prohibit discrimination of victims of domestic violence because of their sexual orientation. Another controversial part of the bill would broaden the authority of Native American tribes to handle cases of domestic violence. 

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Three amendments sponsored by Sens. Kay Bailey Hutchison (R-Texas), John CornynJohn CornynDems confront Kelly after he calls some immigrants 'lazy' McConnell: 'Whoever gets to 60 wins' on immigration GOP senators turning Trump immigration framework into legislation MORE (R-Texas), and Amy KlobucharAmy Jean KlobucharOvernight Regulation: EPA sued over water rule delay | House passes bill to ease ObamaCare calorie rule | Regulators talk bitcoin | Patient groups oppose FDA 'right to try' bill Overnight Finance: Senators near two-year budget deal | Trump would 'love to see a shutdown' over immigration | Dow closes nearly 600 points higher after volatile day | Trade deficit at highest level since 2008 | Pawlenty leaving Wall Street group Dem senator presses FTC to ramp up Equifax hack probe MORE (D-Minn.) would be voted on Thursday afternoon, according to a Senate aide. Hutchison's amendment increases funding for testing backlogged rape kits and keeps the total funding for the act at current levels in an effort to avoid adding to the country's deficit. Likewise Cornyn's amendment increases funding for rape kits and establishes a sexual forensic registry. Klobuchar's amendment also increases funding for rape kits but unlike Cornyn's, does not include a minimum funding requirement. 

After the votes the Senate planned to move immediately to a final vote on the bill. 

The progress to final passage of the bill comes a day after Senate Republicans agreed on Wednesday to allow the legislation to come to a final vote and pass. 

—This story was updated at 4:31 p.m.