The Senate was scheduled to vote on whether attempts to change the Senate rules of procedure would require a majority of 17 votes in the 33-member chamber, or if approval from two-thirds of the members would be needed.
Senate rules determine the powers of the speaker, committees and the flow of legislation.
Republicans held a one-vote majority when the 105th General Assembly convened in January, and Sen. Ron Ramsey of Blountville was elected the first Republican speaker since Reconstruction.
Ramsey appointed Republican chairmen and majorities to the most powerful committees before Sen. Mike Williams of Maynardville shed his Republican affiliation to become an independent.
Williams' defection dropped the GOP into a 16-16 tie with Senate Democrats, but Senate rules do not require Ramsey to change the makeup of committees to reflect that change.
If approved, the two-thirds rule could give Republicans the ability to easily kill any proposals to give Democrats more slots in committees, or even to curb the powers held by the speaker.
Senate Democratic Leader Jim Kyle took the two-thirds requirement for rule changes out of the Senate's temporary rules when the session convened in January. Republicans now want to put the requirement back in before the Senate adopts the permanent rules that would govern the chamber until the end of next year.
Senate Republican Leader Mark Norris said the two-third vote is "in keeping with the trend across the nation that you put protections in place that one can't change the rules in the middle of the game."
Norris, of Collierville, said it would be "wishful thinking" among Democrats that they could cobble together a working majority by gaining Williams' vote.
"Mike Williams is an independent thinker, and he'll do the right thing," Norris said. Williams could not be reached for comment Sunday.
Republicans would need 17 votes to make the tougher rule-change requirements part of the permanent rules. That means they would need either Williams or one Democrat to vote along with them.
Kyle said he wants the Senate to operate under the notion that "the majority should make the rules."
"That is the American way," said Kyle, of Memphis. "All we're asking is that we keep majority rule."
Kyle said another requirement of the rules change provisions supported by the GOP would require all proposals be approved by the five-member Senate Rules Committee. Three of those five are Republicans. Kyle said that rule would "make the entire Senate servant to three members of the Rules Committee, three members of the Republican caucus." Norris said he wants to refer the rules proposals to the committee to prevent possible logjams on the Senate floor. "All these matters need to be referred through the relevant committees, and it's nothing more than that," he said.