Senate Majority Leader Mitch McConnell walks to his office from the Senate floor on Capitol Hill in Washington, US, July 13, 2017. [Photo/Agencies]

"While John is recovering, the Senate will continue our work on legislative items and nominations, and will defer consideration of the Better Care Act," McConnell, R-Ky., said on Saturday. He has already said the Senate will work through the first two weeks of the August recess, citing a need to finish a slate of unfinished business.

McConnell did not indicate when he would aim to return to the health care bill, but Cornyn made clear Sunday that moving quickly is important.

"I believe as soon as we have a full contingent of senators, that we'll have that vote," he said.

Cornyn acknowledged that if the Senate is unable to attract enough GOP votes that it will "keep trying" but will eventually have to come up with a different plan. "We're willing to do what we can to shore up the system now, to stabilize it to make health care available to people now, but we want reforms to go along with it," he said.

The Senate bill, like legislation the House passed earlier, repeals mandates requiring individuals to carry insurance and businesses to offer it, and unravels an expansion of the Medicaid program enacted under President Barack Obama's law. Analyses of the earlier version of the Senate bill found it would result in more than 20 million additional uninsured Americans over a decade compared to current law.

The newest version attempts to attract conservative support by allowing insurers to offer skimpy coverage plans alongside more robust ones, but also reaches out to moderates by adding billions in help for the opioid crisis and to defray high costs for consumers.

In Phoenix, Mayo Clinic Hospital doctors said McCain underwent a "minimally invasive" procedure to remove the nearly 2-inch (5-centimeter) clot and that the surgery went "very well," a hospital statement said. McCain was reported to be resting comfortably at his home in Arizona.

Pathology reports on the clot were expected in the next several days.

McCain, 80, is a three-time survivor of melanoma. Records of his medical exams released in 2008 when he was the GOP candidate for president showed that he has had removed precancerous skin lesions, as well as an early stage squamous cell carcinoma, an easily cured skin cancer.

AP