A close vote had already been predicted for the GOP health care bill, with all Democrats and independents coming out against it, and some Republicans opposed or undecided. With the GOP holding a 52-to-48 majority, they can afford to lose only two Republicans. Vice President Mike Pence would break a tie for final passage.
Two Republicans, Rand Paul of Kentucky and Susan Collins of Maine, have already said they will vote against the measure.
A procedural vote expected in the coming days had been cast as a showdown over the measure designed to replace President Barack Obama's health care law, commonly called Obamacare.
McConnell and other GOP leaders have been urging senators to at least vote in favor of opening debate, which would allow senators to offer amendments. In recent days GOP leaders have expressed optimism that they were getting closer to a version that could pass the Senate.
In Phoenix, Mayo Clinic Hospital doctors said McCain underwent a "minimally invasive" procedure to remove the nearly 2-inch clot and that the surgery went "very well," a hospital statement said. McCain was reported to be resting at his home in Arizona.
McCain 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 precancerous skin lesions removed and had an early stage squamous cell carcinoma, an easily cured skin cancer, removed.