WASHINGTON—President Donald Trump made one of his most explicit threats to cut off payments to insurance companies to force senators and lobbyists back to the bargaining table for a GOP health-care bill, and saying, for the first time, that he was also willing to cancel some of lawmakers’ health-care benefits.

“If a new HealthCare Bill is not approved quickly, BAILOUTS for Insurance Companies and BAILOUTS for Members of Congress will end very soon!” Mr. Trump tweeted Saturday.

Mr. Trump’s statement came less than two days after the collapse in the Senate of Republicans’ effort to overturn the Affordable Care Act.

For months, Mr. Trump has threatened to stop reimbursements to insurance companies—a part of the ACA—but his administration has always paid them in the end, including amid significant uncertainty in June and at a crucial moment in GOP negotiations just a week ago in July.

The next set of payments, which total millions of dollars for insurers that have lowered deductibles and other out-of-pocket costs for the poorest enrollees in coverage under the law also known as Obamacare, is due in three weeks.