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 CornynNew GOP ObamaCare repeal bill gains momentum Week ahead: Senators near deal to stabilize ObamaCare markets GOP eying 'blue slip' break to help Trump fill the courts MORE (R-Texas), and Amy KlobucharAmy Jean KlobucharWeek ahead: Crunch time for defense bill’s cyber reforms | Equifax under scrutiny Some Dems sizzle, others see their stock fall on road to 2020 Consumers the big winners of Amazon-Whole Foods merger 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.