"Our goal is absolutely to make certain that individuals have the opportunity to select their physician," Price told reporters at the White House Tuesday.
It's an example of one of the ways the Trump administration seeks to thread the messaging needle on House GOP efforts to repeal and replace Obamacare.
Price and White House press secretary Sean Spicer both praised the bill during the White House press briefing but stopped short of offering full-throated endorsements, leaving the door open to potential changes to the bill.
Vice President Mike Pence, meanwhile, said plainly that President Donald Trump supports the GOP plan, which Pence helped broker during regular trips to Capitol Hill over the past month.
And President Donald Trump said later Tuesday he was "proud to support the replacement plan released by the House of Representatives" and he hoped to get it passed "very quickly."
The administration's approach reflected the dicey political climate swirling around the proposal, which has already come under fire from conservative advocacy groups and many conservative members of Congress.
Conservative groups and some members of Congress swiftly came out in opposition of the bill, dubbing it "Obamacare-lite" or "Obamacare 2.0."
Price, speaking to reporters during the daily White House briefing, said the bill takes repeal and replace efforts in the "right direction," but made clear it could be subject to changes.
"This is a work in progress and we'll work with the House," Price said, leaving the door open to changes to the current bill.
Spicer declined to say whether it was Trump's bill and said the administration wanted to allow the bill to go through the regular committee process, which would make it subject to amendments.
"We're not jamming it down anyone's throats," Spicer said, accusing the Obama administration of doing just that to pass the Affordable Care Act.
Pence, who spent the day meeting with Hill Republicans, said following the GOP Senate lunch that Trump supports the plan. But he acknowledged the White House was open to changes in the measure.
"As the legislative process goes forward, the President and I believe that the American Health Care Act is the framework for reform. We're certainly open to improvements and recommendations in the legislative process," Pence said. "But this is the bill, and the President supports the American Health Care Act and looks forward to working very directly with the leadership of the House and Senate to move this bill."
Spicer made clear that the administration is invested in the bill, noting the administration has worked closely with Congress to craft the legislation and that the White House's legislative staff was encouraging members to support the bill.
Pitching the proposal from the White House briefing room, Price also avoided making any promises about the House GOP bill, like whether Americans would be able to keep their doctor as the Obama administration erroneously promised in selling Obamacare and could not guarantee that Americans won't lose their health care coverage.
"Those promises turned out to be not true," Price said. "Our goal is absolutely to make certain that individuals have the opportunity to select their physician."
He also said he "strongly" believes the House GOP bill will lead to a decrease in premium costs and the cost of health care overall, but stopped short of issuing any guarantees. And Price also declined to guarantee that Americans would not see any coverage losses.