A Kentucky bill that would allow Sen. Rand PaulRandal (Rand) Howard PaulSenate Dems hold floor talk-a-thon against latest ObamaCare repeal bill Overnight Defense: Senate passes 0B defense bill | 3,000 US troops heading to Afghanistan | Two more Navy officials fired over ship collisions Lawmakers grapple with warrantless wiretapping program MORE (R-Ky.) to run for the Senate and White House at the same time passed the state Senate on Tuesday night. 

The proposal advanced with all Republicans voting in favor of it, except one. Two Democrats voted for it, according to The Lexington Herald-Leader

However, House Speaker Greg Stumbo (D) has consistently said the bill likely will not advance in the Democratically controlled House. 

"We thank the Kentucky Senate for recognizing the need to clarify the law to avoid a conflict with the U.S. Constitution," Paul adviser Doug Stafford told the newspaper. 

Paul's camp, which has pushed for the proposal, has maintained an earlier Supreme Court ruling has already found state law does not govern federal elections. Paul is seen as a likely contender for the GOP nomination in 2016. 

Kentucky Sen. Damon Thayer (R) earlier this month introduced the bill, which is aimed at clarifying a state law that says no candidate's name can appear on a ballot more than once in the state. The bill would clarify that the law does not apply to federal elections. 

He said Paul is one of a dozen people in the country who could realistically become the next president and did not want to "deny the people of Kentucky an opportunity to vote for or against one of their own."

A recent CNN poll showed Paul narrowly leading the pack for the GOP nomination with 16 percent of the vote. He is also coming off a string of presidential straw poll wins in New Hampshire and the Conservative Political Action Conference.