Sen. Lamar Alexander, R-Tenn., said Thursday that "a significant number" of both Democrat and Republican co-sponsors had been added to his bipartisan bill to stabilize Obamacare, despite opposition to the bill by President Trump and House Speaker Paul Ryan.

Alexander said during a Health, Education, Labor and Pensions Committee hearing, which he chairs, that he spoke with Trump Wednesday night, and said Trump offered some suggestions of how the bill, which he now opposes, could earn his support.

"He encouraged the process, which he asked me to begin, and said he looked forward to considering it," Alexander said. "I said, 'If you have suggestions for improving it that's certainly your prerogative to do, and that's what we would expect to happen in the legislative process.'"

Alexander said he will appear on the Senate floor at 1 p.m. Thursday with Sen. Patty Murray, the committee's top-ranking Democrat, whom he called a "straightforward, tough negotiating partner," in support of the bill.

Alexander and Murray's bill would fund cost-sharing reduction payments for insurers for the rest of the year and for two additional years, and would allow more people to purchase "copper" plans with lower premiums and higher deductibles. It also would accelerate state's ability to obtain waivers for how they implement Obamacare.

The White House has spelled out Trump's suggestions to the bill, including that it provide "relief" from the individual and employer mandates and that it allow more people to band together to form "association health plans," which would be similar to an executive order he signed a week ago.

Conservatives have said that the package is a "bailout" to health insurance companies and say that it does not fulfill Republicans' seven-year promise to repeal and replace the law.

Alexander expressed relief at changing the topic at the HELP Committee hearing, which was about what could be done to encourage people to make healthier choices to help prevent serious illness and to reduce healthcare costs.

"I am delighted to be talking about anything other than the health insurance market," he said. "I'm really quite serious about that. We know that the largest issues in healthcare are much more than the 6 percent of people, every one of who is important, who have to buy insurance through the individual health insurance market."