U.S. Senator Lindsey Graham (R-SC) and Senator Chuck Schumer (D-NY) (R) speak during a push for new bipartisan media shield legislation during a news conference at the U.S. Capitol in Washington, July 17, 2013. Also pictured is co-sponsor Senator Amy Klobuchar (D-MN) (L). REUTERS/Jonathan Ernst (UNITED STATES - Tags: POLITICS MEDIA) Sen. Lindsey Graham (R-S.C.)  (Jonathan Ernst/Reuters)

Senator Lindsey Graham (R-S.C.) called on Senate leaders to allow a vote before the midterm elections on a proposal that would ban abortions after 20 weeks, even if it might not have the support to pass.

"I'm asking for Senator McConnell and Senator Reid to allow us to vote on this, in 2014. I know we are really busy around here and there's no room to do anything, but we'll find some time to talk about this issue," he said Thursday during a news conference with Sen. Rob Portman (R-Ohio) and several other members of the antiabortion community.

While Graham conceded that there would be "short of 50" votes for the "Pain-Capable Unborn Child Protection Act," he said support would grow as the public better understood the legislation.

Graham's bill would ban abortions starting at 20 weeks of gestation except in cases of rape, "incest against a minor" or when the mother's life is in danger.

A similar bill passed the Republican-controlled House in June.

Graham's proposal marks the second time in as many weeks that the South Carolinian has publically pushed issues popular with the right.

Last week, Graham pledged to block executive nominations until Congress was able to interview survivors of the attack last year on the U.S. consulate in Benghazi.

Graham, who faces a primary challenge from the right next summer, brushed away a question about the timing of the proposal that has little chance of going anywhere in the Democratically controlled Senate.

"I came into the political arena pro-life, and I will leave pro-life," he said. "Now whether or not I leave next June or some later period, will be up to the voters in South Carolina."

The right blasted Graham as recently as last month for his refusal to support a proposal to tie the debt ceiling and government shutdown to defunding the Affordable Care Act.

On Thursday, Graham said this most recent proposal would likely do little to erase other issues with his record.

"Did I wake up because I got a primary and say, 'Hey, let's be pro-life?' No, I'm honored to do this, this is important to me, this is why I want to be a senator, " he said. "Will it wipe away all the other criticisms?  No."