Graham and Cassidy, who have been working closely with former Sen. Rick Santorum, argue that one of the main reasons that Republicans are having such a hard time agreeing is because they are working from the Obamacare template -- particularly federal control of health insurance.
"The reason we can't pass a bill is because we are trying to do it in Washington, so stop it," Santorum, a CNN contributor, told CNN.
Additionally, the version of the Republican bill first unveiled last month has turned out to be a tough sell back home because it repeals Obamacare taxes, effectively giving wealthier Americans a tax cut, while also making cuts to Medicaid, which now helps many afford health insurance.
At the same time that Graham is unveiling his plan, GOP Senate leaders are expected to release publicly a new version of their proposal Thursday morning, though it's unclear what exactly will be included in the new bill or whether those changes to the original will bring on board additional votes.
Graham told CNN that he believe their new idea would address the problems found in the original bill.
First, it would maintain taxes on wealthy Americans put in place by Obamacare, taxes intended to help pay for insurance subsidies. That revenue, which they estimate to be about $500 billion, would be sent to the states.They are still waiting to see a score from the Congressional Budget Office to confirm that figure.
The Obamacare tax levied on medical device makers would be repealed, as would the financial penalty for individuals who don't comply with the mandate to have insurance and for employers to provide affordable coverage. Graham and Santorum say that would amount to about $220 billion in revenue cuts. Graham and Santorum argue those tax penalties not only funded the Affordable Care Act mandate conservatives despise, but they also ended up hitting working people.
The idea is modeled on the Welfare Reform Act of 1996, which dismantled the federal entitlement program with block grants to the states.
That, however, was done in a bipartisan way -- with GOP leaders like Santorum working with then-President Bill Clinton.
For the latest health care discussions, no Democrats have signed onto the legislation and it is unlikely many, if any, would even vote for such an idea.
Instead, the name of the game is finding elusive consensus just among Republicans -- and even that may not be possible.
Graham and Santorum say the key to getting traction on their idea is buy in from GOP governors, some of whom have significant influence with their Republican senators.
One of the aspects of this proposal would be to include so-called "inflators" -- which would allow states that find formulas for quality, affordable health care to keep the federal dollars they save by doing so.
Graham and Santorum both say they have been talking to one another seriously about this idea for months, but when the Senate GOP bill began to falter at the end of June, they kicked it into high gear.
Santorum went to the White House last week to discuss the idea with senior aides to President Donald Trump, and he and Graham have been lobbying Senate colleagues, seeking input.
A key unanswered question in this proposal is what happens to the essential health benefits
mandated under Obamacare.
One that Graham says will not go away is the popular ban on denying coverage for pre-existing conditions, which congressional Republicans and the President promised to keep.