"What I'm telling you is the system, the plan that we would put in place, would not allow individuals to fall through the cracks", Price told CNN's Dana Bash.
President Trump spent part of his Saturday at Trump National Golf Club but had healthcare on his mind.
Trump's role is expected to become more pronounced in coming days as the vote nears. Senate Majority leader Mitch McConnell smiles as he leaves the chamber after announcing the release of the Republicans' healthcare bill. "They are living healthier and happier lives because of that decision - I don' think that can be overstated enough".
That could add Heller's name to Trump's call list.
He said he has made his views clear to the party leadership and the White House.
The claim that the bill would lower consumers' insurance premiums was "a lie", he said.
Health care stocks pulled back Friday, but the sector was still on track to post a weekly gain of more than 3 percent.
The debate underscores the partisan divide in overhauling ObamaCare - former President Barack Obama's signature, 2010 health care law that extends health coverage to millions more Americans but has also struggled with rising premium costs and fewer premium options. She said that a loss in federal funding could be detrimental to patients who rely on health care services including dentistry, OB/GYN, and pediatric medicine. Some Republican senators say the bill doesn't go far enough and are calling for a complete repeal.
CBO analysis of the bill predicts that 23 million Americans would lose their health insurance, while the top 1 percent of Americans would enjoy $306 billion in tax cuts over 10 years.
While discussing the Republican health bill during an interview that aired Sunday on Fox & Friends, Trump said it would be great if lawmakers from both parties could "come up something that everybody's happy with".
"I want to get to yes, and the way to get to yes is to fix the underlying problems". "We'll have to see".
Trump said getting approval would require travelling a "very, very narrow path" but that "I think we're going to get there".
The draft bill comes nearly two months after the House passed its own measure.
No Senate Democrat backs the new GOP plan. "Well, I've done in five months what other people haven't done in years", he said.
Johnson identified his opinion as "not a yes yet" on the Senate healthcare bill.