WASHINGTON — Despite a rushed timetable and the daunting task of pulling together enough votes, the White House is “planning” on the Senate voting on the “Graham-Cassidy” health care bill this week, White House Director of Legislative Affairs Marc Short said on Sunday’s “Meet The Press.”

He claims they are aiming for Wednesday.

“There are millions of Americans who will benefit from this bill,” Short said. “In fact, we think every state will benefit.”

Majority Leader Mitch McConnell, who decides when the Senate could address the bill, has said that this week they “intend” to consider the legislation that’s aimed at restructuring the U.S. health care system and repealing some of the signature elements of the Affordable Care Act, also known as Obamacare.

But doubts remain over whether he would bring up the bill if it did not have enough support to pass, especially after Republicans’ last attempt at repealing Obamacare suffered a theatrical failure on the floor in July.

The White House and Senate Republican leaders believe that Graham-Cassidy is their last shot at dismantling Obamacare this year, since their deadline to pass something without a 60-vote majority is at the end of the month.

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The bill’s prospects, however, were dealt a major blow on Friday when Sen. John McCain, R-Ariz., announced that he could not “in good conscience” vote for it. If the legislation loses support from three Republicans, it will not pass, since it does not have the support of any Democrat.

Short dodged the question on Sunday when specifically asked which votes were in play.

Sen. Lisa Murkowsi, R-Alaska, has not firmly said whether she would support the bill – claiming she wants to see more information about how the legislation could impact the health care markets in her state. Sen. Susan Collins, R-Maine, told CNN on Sunday that it is “very difficult” to imagine a scenario where she backs the bill.

Sen. Rand Paul, R-Ky., who has been one of the bill’s fiercest Republican critics, set a bar for his support so high that it’s likely impossible the legislation could meet it.

"This is a bad idea, it’s not repeal," Paul said in an exclusive interview on Sunday’s "Meet The Press."

"The Graham-Cassidy bill basically keeps most of the Obamacare spending, almost all of the spending, and just reshuffles and block grants it to the states," he added. "So, block granting Obamacare doesn't make it go away. It just means you're keeping all the money we've spending through Obamacare, most of it, reshuffling it, taking the money from Democrat states and giving it to Republican states. I think what it sets up is a perpetual food fight over the formula."

Paul called the suggestion that he opposes the bill because he actually doesn’t want to see major changes to Obamacare — since a number of people from his home state of Kentucky are satisfied with it — a “personal insult” to his character.

"I think most people who follow my position would find that as a physician, there's not been a more consistent voice for repealing Obamacare," he said.

The senator added that if they could "remove the block grants" from the bill, or "block grant at pre-Obama levels," he could vote for it. But those block grants are essential to what’s inside Graham-Cassidy.

Graham-Cassidy removes both Obamacare’s mandate that individuals must purchase insurance, as well as the mandate for some employers to offer insurance plans to their employees. It takes funds that would have gone to the Medicaid expansion under Obamacare and instead introduces block grants for states to decide how they want their own health insurance systems to work.

Short meanwhile on Sunday continued to claim that under Graham-Cassidy, people with pre-existing conditions would be "protected," although in the current version of the bill, that is not as straightforward.

The requirements would be loosened compared to what is currently law under the Affordable Care Act.

Under Graham-Cassidy’s language, a state must show how it "intends to maintain access to adequate and affordable health insurance coverage for individuals with pre-existing conditions" if it requests a waiver. However, it offers no additional guidance or details on how that would be binding, leaving major questions for coverage for people with pre-existing conditions.