The Republican senators sponsoring the latest bill aimed at overhauling the nation's health care system have made revisions that would send more money to several states represented by senators whose support is needed to pass the measure.
A chart Republicans circulated Sunday said the updated version of the bill would boost grants to Arizona, Kentucky, Texas and Alaska.
Republicans hold a 52-48 majority in the Senate, and with no Democrats expected to support the overhaul effort, if three Republicans vote against it, the legislation will fail.
Arizona Sen. John McCain and Kentucky Sen. Rand Paul have said they will not support the bill.
Texas Sen. Ted Cruz expressed opposition to the bill Sunday, saying, "Right now, they don't have my vote." He added that he does not think Utah Sen. Mike Lee was ready to back the bill either.
Opponents include those who think the overhaul effort does not do enough to bring down insurance premiums or cuts too much from programs to help insure the poor, and those who say it does not cut enough and instead leaves in place too much of the program championed by former President Barack Obama.
Sen. Lisa Murkowski of Alaska has also voiced reservations about the plan to repeal the Affordable Care Act, often referred to as Obamacare.
Sen. Susan Collins of Main told CNN on Sunday she finds it "very difficult" to imagine herself voting to upend the ACA, but that she was not ready to declare with certainty she would vote no without first seeing an assessment of its impact by the independent Congressional Budget Office.
The latest repeal effort is being advanced by Senators Lindsay Graham of South Carolina and Bill Cassidy of Louisiana. It would send block grants of federal money to the country's 50 states and allow them to shape independent health care plans for the poorest people in their states, rather than the current system of having the federal government allocate the money.
Collins has expressed the view that millions of people who now have insurance to pay their health care bills under Obamacare would lose it under the Republican plan. National opinion surveys show that overall a majority of Americans want to keep the current law, although many Republicans do not.
Graham told ABC News that he still thinks there is a possibility of winning the repeal vote and is "moving forward" with efforts to try to change the minds of the last Republican holdouts against the legislation.
Republican Senate leader Mitch McConnell says the Senate will consider the repeal effort this week, just days ahead of a September 30 deadline, as dictated by the Senate's legislative rules, for completing action.
Even so, it is not clear there will be a vote because McConnell has said he will only call for a vote if he is assured of having enough votes to repeal the law. If the Senate repeals the law, the House of Representatives would then take up the repeal effort.
Numerous Democratic critics of the proposal, along with several major health care groups in the country, say that over the coming years the change in the law would cut millions of people from insurance rolls.
Millions of Americans have been able to secure insurance to help pay their health care costs under Obamacare, but Republicans oppose it because of the mandate to buy insurance coverage and provisions in the law that have led to big increases in the cost of insurance policies for individuals who do not have coverage through their employers, as millions of American workers do.
U.S. President Donald Trump, anxious for a major legislative victory, has called for the repeal, saying in a Twitter comment last week, "I hope Republican Senators will vote for Graham-Cassidy and fulfill their promise to Repeal & Replace ObamaCare. Money direct to States!"
Obama has only rarely publicly engaged in the day-to-day Washington policy fights since Trump replaced him in January.
But last week Obama attacked Republicans for trying to overturn the health care law.
“The legislation that we passed was full of things that still need to be fixed; it wasn’t perfect, but it was better." Obama said. "And so, when I see people trying to undue that hard-won progress for the 50th or 60th time with bills that would raise costs or reduce coverage or roll back protections for older Americans or people with pre-existing conditions, the cancer survivor, the expecting mom, the child with autism or asthma for whom coverage, once again, would be almost unattainable."
"It is aggravating," Obama said. "And all of this being done without any demonstrable economic or actuarial or plain common-sense rationale — it frustrates. And it certainly is frustrating to have to mobilize every couple of months to keep our leaders from inflicting real human suffering on our constituents."