Sen. Susan Collins, R-Maine, a moderate who also has concerns about the GOP bill and whose vote will be critical, said the Senate will "start from scratch" and not consider the House bill.
Another target is Rep. Dave Brat of Virginia, a member of the House Freedom Caucus, the group of conservative Republicans whose objections to an earlier version of the bill helped scuttle a House vote scheduled for late March.
The House GOP did not have the final version of their bill scored last week, but it was reviewed in prior forms, with the CBO concluding that 24 million more Americans would lack health insurance by 2026 if the previous bill became law.
The Senate's different approach means there's no clear timetable for producing a bill, and it likely ensures that President Donald Trump and House Republicans will eventually have to face legislation that doesn't fully repeal the Affordable Care Act despite their repeated campaign promises to do it.
"We're going to draft our own bill and I'm convinced that we're going to take the time to do it right", she said.
"What we're trying to do is to make certain that every single person has health coverage", he said.
Democrats are also refusing to participate in any effort to dismantle Mr Obama's law, while some Republican senators object to cutting Medicaid, the federal-state health care programme for the poor and disabled.
Price said during appearances on two Sunday talk shows that Medicaid was fundamentally flawed. But Republicans insisted their office seats were made possible, in many cases, by constituents who have implored them to end Obamacare and improve health care coverage.
The House bill would eliminate fines on people who don't buy policies and erase taxes on health industry businesses and higher earners. It would allow states to seek waivers allowing them to deny coverage to people with pre-existing conditions.
The Vermont Medical Society (VMS) opposes yesterday's passage of the American Health Care Act (AHCA) in the United States House of Representatives.
However, he also seemed to make a point of giving the Senate time to make changes, despite Trump's strong desire to fulfill his campaign promise of repealing and replacing ObamaCare.
One proposal from a Republican congressman may have swayed some undecided members in favor of the bill - the amendment would add $8 billion over five years to cover insurance for people with pre-existing conditions.
The future of what many are calling "Trumpcare" could hinge on 7 Republican Senators who have voiced objections on similar legislation in the past.
House Republicans' message to the American people?
In his Twitter comment, Trump contended, "ObamaCare premiums and deductibles are way up - it was a lie and it is dead!"
"Recent changes did not address fundamental concerns about the bill's impact on the 650,000 individuals that are part of our Medicaid expansion population", he said, "nor have those changes eased the concerns of the 350,000 people in the individual market who are dealing with skyrocketing premiums and fewer choices". "The Senate will complete the job".