"At the moment it looks like everything has stalled out," Thune said
. "So we'll see where it goes from here, if there's a will to kind of put it together and see if there's a path forward."
In the Senate, many Republican rank-and-file members said they had a lot more work to do before they would be ready to support Alexander-Murray.
"I'm studying it," said Sen. John Cornyn, a Republican from Texas. "I think we have a lot of discussing to do. It was just introduced."
Behind the scenes, however, the bill's sponsors Sens. Lamar Alexander, a Republican from Tennessee, and Patty Murray, a Democrat from Washington, are still working to build consensus for their legislation that would fund what are known as cost-sharing reduction payments
for two years in exchange for giving states more flexibility in what insurance plans can be offered in the Obamacare marketplace. The President announced last week that he would no longer make the CSR payments, which help low-income Americans cover their out-of-pocket insurance costs.
The plan for now is to play the long game on the bill. Alexander and Murray -- both experienced bipartisan negotiators in the Senate -- may have seen setbacks, but neither one is prepared to step away, aides say.
An Alexander aide told CNN Tuesday that Murray and Alexander could announce as many as 20 co-sponsors -- 10 Republicans and 10 Democrats -- as early as Thursday in an attempt to reignite momentum for the bill. And while Alexander and Murray worked for weeks to carefully craft their compromise, aides say that the senators are still open now to making tweaks.
"Nothing is final until it's actually in a car driving down Pennsylvania Avenue for a signature. It would be naïve to say nothing will change. It would be naïve to say it will be a complete rewrite," an Alexander aide said when asked about changes to the legislation.
A Democratic aide told CNN: "There are already strong provision in the agreement that make clear insurers cannot double dip, or both receive payments to reduce out-of-pocket costs for consumers and charge higher premiums because they'd factored in President Trump's decision to cut those payments off. If Republicans are interested in other ways to make sure patients come before insurance companies in the agreement, Democrats are certainly more than open to that."
For now, many of the hurdles are on the Republican side. For one, Alexander is fighting years' worth of rhetoric surrounding the CSR payments. The Alexander aide said that when the White House press secretary called the CSR payments an insurance bailout in the White House briefing it was "not helpful."
"We don't want an insurance company bailout ... Patty Murray doesn't want an insurance bailout," the aide said.
During the Obama administration and up until last Thursday when Trump announced he'd no longer make the payments, the CSRs were made by the administration. That move was the subject of a House lawsuit charging the administration did not have the authority to make the payments without congressional approval.
On the one hand, some rank-and-file Republicans argue that the payments should be appropriated by Congress in order to keep insurers in the marketplace and make sure that insurance premiums don't rise in the future. But, it's a tough political pill to swallow. The GOP, after all, has spent years railing against the Affordable Care Act. Working now to stabilize it is a hard sell and members of the House including Speaker Paul Ryan have voiced strong concerns with the Alexander-Murray bill.
The focus in the meantime is to keep educating members on what will happen without the payments. Under the law, insurers can pull out of the marketplace and health care experts have warned that the government could end up paying even more to ensure low-income people in the future without the payments.
It's still unclear how Alexander and Murray would move their bill. There is little time left on the Senate calendar as Majority Leader Mitch McConnell will be focused on tax reform and a must-pass spending bill in upcoming months. However, senators who support Alexander-Murray say the aim is to keep building support behind the scenes and hope that at some point, Alexander-Murray can be attached to a moving vehicle.
"At some point, it's going to happen. I think we all know that," said Sen. Bob Corker, a Republican from Tennessee and a co-sponsor of the bill. "At some point, it's going to be attached to something right?"
"Again, I want to say one more time. The President called Senator Alexander twice asking him to pass this piece of legislation," Corker added.