Senate Republicans expect to release a revised version of a bill to roll back the 2010 health care law this week and vote on it the following week, Sen. John Cornyn, R-Texas, told reporters on Monday.

The lawmakers are considering several changes to a draft bill that Republicans released in June after weeks of closed-door meetings that nearly a dozen Republicans have said they don’t support. GOP leaders can afford to only lose two votes on a health care bill, which would allow Vice President Mike Pence to cast a tie-breaking vote.

GOP leaders have sent two versions of revised legislation to the Congressional Budget Office for review.

In addition to the 10 Republicans who have said they wouldn’t vote for the current bill, others have raised concerns and withheld their support. It would be an uphill climb for Republican leaders to pass the measure before the August recess.

In a nod to the challenge at hand, Republicans have already begun talking about what to do if the Senate fails to pass an overhaul.

Many GOP senators, including Majority Leader Mitch McConnell of Kentucky, have suggested the next step should be to pass a bipartisan measure that stabilizes the individual insurance markets.

But the White House and conservatives are pushing for a broad repeal bill if Senate Republicans can’t reach an agreement on a replacement plan.