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(CNN) —  

As Republicans on Capitol Hill grapple with President Donald Trump’s decision to thrust health care back onto their agenda, many are looking to the White House for guidance on a plan. But the White House doesn’t have one, and aides are suggesting it could be up to Congress to determine what would replace the Affordable Care Act should the courts strike down the law.

The disconnect over who is primarily responsible for drafting a replacement plan underscores how unpopular the President’s decision is and how difficult the path ahead could be for a party that failed to pass health care legislation when they controlled all three branches of government – a monopoly Republicans no longer enjoy.

Marc Short, chief of staff to Vice President Mike Pence, said Wednesday on CNN that “the President will be putting plans forward this year” on a health care replacement plan.

But a White House official said Thursday that Short “spoke too soon” because aides are still debating whether the administration will take the lead in crafting a proposal or ask Congress to step in. The official said White House aides are looking at Graham-Cassidy as their starting point, and could either make changes to it themselves or work with Congress on a proposal.

Whether the White House or Congress should take lead is “still being decided,” the official said, among other issues officials are working out in senior staff meetings on Thursday.

It’s not like any suggestions will get far, though, with a Democratically-controlled House.

Trump told the attendees of the Senate GOP lunch on Tuesday that Congress needs to come up with a better law than the Affordable Care Act, and he has repeatedly hyped the GOP as the “party of health care,” despite that party’s reluctance to engage the issue.

But he told reporters Thursday that he is in “no very great rush” to come up with an Obamacare alternative.

“There’s no very great rush from the standpoint that we’re waiting for the decisions from the court,” Trump said before departing for a Michigan rally, adding that the White House has put together a team of Republican senators, including Bill Cassidy, Rick Scott and John Barrasso, to “form a really great plan.”

Meanwhile, some Republicans are looking to the White House to provide them with a plan.

“I’m anxious to see what the White House is going to recommend in terms of a health care delivery system that looks like somebody designed the damn thing on purpose,” said Sen. John Kennedy, R-Louisiana.

Sen. Roy Blunt, R-Missouri, responded to questions about the health care plan: “All I’m going to say about that for a while is that I’m eager to see what the administration proposes.”

“The President, obviously, has something in mind. We’ll wait and see what it is,” said Sen. Lamar Alexander, R-Tennessee, chairman of the Senate Health, Education, Labor and Pensions Committee.

Sen. John Cornyn, a member of Republican leadership, said: “We are waiting with bated breath” for the White House’s proposal on a new health care plan.

Many GOP lawmakers feel the White House should own the health care issue because the President is the one who dragged Republicans back into a battle they lost.

One Republican congressman told CNN on Thursday that after Trump received a political boost from the end of the special counsel probe this week, he spiked the football in the end zone and the “damn ball came up and hit us in the nose.”

CNN’s Ted Barrett, Lauren Fox and Jeremy Diamond contributed to this report.