WASHINGTON—Senate Majority Leader Mitch McConnell, ratcheting up the pressure on fellow Republicans, set a vote on the GOP health-care bill for next week and delayed the chamber’s August recess to give the party time to seek other legislative wins, even if the health plan stalls.

Senate GOP leaders, still working to corral support for the health legislation, are now expected to retain the Affordable Care Act’s two taxes on high-income households, lawmakers and aides said Tuesday—an unusual move for the tax-averse GOP.

But while that would give Republicans more money to sweeten the bill to win over GOP holdouts, 10 Senate Republicans still remain publicly opposed to the legislation. With Democrats unlikely to support the bill, Mr. McConnell (R., Ky.) can lose no more than two GOP votes to pass it. Vice President Mike Pence could break a potential 50-50 tie.

“We’re going to do health care next week,” Mr. McConnell told reporters Tuesday. “And then we’re going to turn to other issues,” he said, citing the annual defense policy bill, confirmation of President Donald Trump’s nominees and raising the federal government’s borrowing limit among the summer’s goals.

Such a strategy could help Senate Republicans avoid returning to their home states empty-handed in August if the health bill flops, an outcome that would likely anger Mr. Trump, conservative voters and House Republicans, who passed their version of the bill in May.