Addressing a press conference in the Rose Garden with President Trump, McConnell said, "It's supported by virtually all of my members, and we expect to have a vote on it at some point."
He, however, didn't specify when the legislation will come up.
The House on October 3 passed a bill to make abortion after 20 weeks illegal in every state in the country. Called the Pain-Capable Unborn Child Protection Act, it is based on the idea that a fetus at 20 weeks' gestation can feel pain.
The bill bans abortions after 20 weeks, except in cases of rape, incest, or a threat to the life of the mother. A doctor who performed an abortion after 20 weeks, except in those cases, could face up to five years in prison. Women seeking abortions would not be penalized under the bill.
With a 52-seat majority, Republicans are unlikely to be able to pass the bill in the Senate. They would need 60 votes, including the support of at least eight Democrats, to clear the legislation, reports The Hill.
Graham's bill currently has 45 cosponsors, none of whom are Democrats. It will be eligible for a vote through 2018, when the current Congress ends.