Senator John Cornyn now says he supports a talking filibuster.
His New York Post Op-Ed comes after some republicans have called for rules changes to ultimately pass voting legislation.
"After careful consideration, I support whatever changes to Senate rules that may prove necessary for us to get the SAVE America Act and homeland security funding past the Democrats' obstruction, through the Senate, and on the president's desk for his signature," Cornyn wrote.
The SAVE America Act would require U.S. citizenship to register to vote and voter ID at the polls.
"This could be a 'talking filibuster' that removes the obstructionists' free pass and makes them defend their indefensible views on the Senate floor," Cornyn wrote.
Democrats have argued the SAVE American Act would make it more difficult for some Americans to vote.
President Donald Trump had indicated over the weekend that he could refuse to sign any bills until the SAVE America Act is passed.
"I respectfully urge the remaining handful of my Republican colleagues still holding on to the old position that I used to share to reassess the new reality and update their thinking," Cornyn wrote.