Senate GOP leaders are planning to vote next week on legislation to repeal large chunks of the Affordable Care Act, even though they don’t yet appear to have secured enough support to pass it.

Senate Majority Leader Mitch McConnell (R., Ky.) is intent on keeping pressure on Senate Republicans to move quickly on the bill rolling back and replacing much of the 2010 health law, lawmakers and GOP aides said. The push for a quick vote could backfire, however, as some conservative and centrist Republicans have expressed concern about the emerging shape of the bill.

Mr. McConnell could pull back if he calculates that a little extra time could get him the votes needed to cross the finish line. He can lose no more than two GOP votes for a bill to pass. All Democratic senators are expected to oppose it.

Still, Mr. McConnell has reasons to try to vote quickly on the health-care bill: the pressure could force lawmakers to reach a consensus on sticking points that have divided them for weeks. And GOP leaders in both chambers want to move on to other items on their legislative agenda, including a sweeping overhaul of the tax code.