The President has talked up the possibility of a 10% tax cut for middle-income Americans on the campaign trail and in the White House in recent days. But the idea, which appeared to materialize out of thin air, has taken Republicans on Capitol Hill by surprise and prompted Trump's own economic adviser on Tuesday to concede that a ny cuts "may not surface for a while."

When Trump first floated the idea of a new tax cut over the weekend, he suggested it could be unveiled ahead of the midterms. The House and Senate are out of session through the midterm elections, however, which are now just two weeks away.

Beyond that, it's not clear there is any specific legislative proposal on Capitol Hill that would achieve the President's goal. And even if Republicans push for tax cuts after the midterms, it could be difficult, if not impossible, for any new tax cut to pass both chambers when lawmakers return to Washington for a lame-duck session before a new Congress takes over in January.

When Trump debuted his new talking point, it sent aides on Capitol Hill spinning to figure out what he meant, and in the House it sent them rushing to figure out if they could -- or even need to -- draft something that would address what the President was talking about. In the Senate, GOP officials said calls and emails were sent to their House counterparts for guidance this past weekend, only to find out there wasn't any -- nobody was sure what exactly the President was referencing.

White House officials, too, were caught off guard by the President's remarks. Some officials on the economic and legislative teams spent the weekend and early part of the week attempting to figure out where Trump got his new pitch and fielding confused phone calls from allies on Capitol Hill, according to two sources familiar with the internal reaction. There was also the open question of whether there was actual work to be done in formulating a new proposal or, as one of the sources put it, "it was just something POTUS felt like tossing into the conversation in the weeks leading up the election."

To be sure, in a Republican-controlled House, legislative proposals can move quickly if prioritized by leadership, including bypassing the committee process altogether. But there are currently no plans to do anything of the sort, aides said -- primarily because nobody has pinned down what, exactly, the President is talking about.

"Your guess is as good as mine," one senior House GOP aide said.

As to whether something could eventually happen?

"I guess," the aide said. "But it's not like we don't have a lot on our plate after the election."

Key Republican chairman speaks up

On Tuesday, Rep. Kevin Brady, a Texas Republican who's the chairman of the powerful House Ways and Means Committee, the chamber's tax-writing body, said he will work with the Trump administration to come up with a plan in line with what the President has suggested, though he didn't offer any guarantees on whether it would pass.

"We will continue to work with the White House and Treasury over the coming weeks to develop an additional 10 percent tax cut focused specifically on middle-class families and workers, to be advanced as Republicans retain the House and Senate," Brady said in a statement.

Trump cited the Ways and Means chairman in comments he made at the White House on Tuesday afternoon that Republicans would unveil details of the plan in the coming days.

"It's going to be a resolution, probably introduced this week, the end of the week or early next week," he said. "And Kevin Brady has been drawing it up, actually, for a while. We've been working on it very hard for a pretty long period of time."

One congressional leadership aide told CNN before Trump's comments Tuesday that details are fuzzy and reiterated a slower timeline than Trump has promised in recent days, adding, "No one knows how much it costs."

One potential obstacle for any new plan is that the House will be primarily focused on two things when it returns in November: the fallout from the election and funding the government

It is unlikely that any new tax proposal the President comes up with would win Democratic support, especially if Democrats win control of the House in the midterms. And why House GOP leaders -- including retiring Speaker Paul Ryan of Wisconsin -- would want to take up a new tax cut with no future in the Senate, as leadership battles are playing out and dozens of retiring or defeated members are heading for the exits , is an unanswered question.

'Building on the success of the Tax Cuts and Jobs Act'

House Republicans have already done their election year politics tax cut -- it was the "Tax Reform 2.0" package the chamber passed in September before recessing until after the election.

The House Ways and Means Committee moved that package of tax reductions through the legislative process. No such work has been done on Trump's new cuts -- in fact, no work on Capitol Hill had been done at all on a new tax reduction proposal of any kind, 10% or otherwise, senior GOP aides told CNN. That's because they weren't aware the President would call for a new proposal, the aides said.

Asked for details of Trump's proposal at the start of the week, spokespeople for the speaker's office and the Ways and Means Committee initially referred all questions to the White House.

"I can tell you there is continued interest in building on the success of the Tax Cuts and Jobs Act and constantly improving the tax code for hardworking families and America's small businesses," Ways and Means Committee spokesman Rob Damschen said later in a statement. A spokeswoman for Ryan referred questions from CNN to the White House.

Trump acknowledged on Monday that there would not be time for Congress to vote through new tax cuts before the November 6 midterms. "We'll do the vote later. We'll do the vote after the election," he said.

When the House passed its tax package in September -- which included permanently extending the 2017 tax cuts for individuals -- House GOP aides made abundantly clear what they were doing: trying to give their endangered incumbents another issue to campaign on.

The reason? There was no chance the Senate would take up the package -- it simply didn't have the votes. Unlike the 2017 tax overhaul, which needed only a simple majority to pass the chamber due to the use of an arcane budget process as its legislative vehicle, the second round would need 60 votes to move forward under Senate rules. Senate aides in both parties said the bill would fall far short of that number.

But despite the fact that the Senate is not expected to take up the House's pre-midterm tax package because it does not have the votes needed to pass, White House spokeswoman Lindsay Walters suggested on Monday that the President would like to see his tax cut for the middle class included as part of that package.

"As part of Tax Reform 2.0, the first elements of which ... passed the House in September, the President would like to see an additional tax cut of 10% for middle-income families," she said in a statement.

Only three House Democrats supported the second round of tax cuts, two of whom -- Reps. Jacky Rosen of Nevada and Kyrsten Sinema of Arizona -- are running in tough campaigns for Senate seats. The third was Rep. Conor Lamb, a moderate Democrat running in a Pennsylvania district that leans Republican.

'It's a talking point'

That the effort undertaken was both known to be futile legislatively and proposed as a political tool for House Republicans goes to show just how far out of left field Trump's new talking point is on Capitol Hill. He's essentially asking for the same thing -- but without any legislative work backing his new idea and amid a deficit shown to be ballooning.

Kyle Pomerleau, an economist at the Tax Foundation, a nonprofit think tank in Washington, said it would be highly unlikely for the President's talked-about cut to pass either prior to the midterms or during the lame-duck period.

"We don't have details of any proposal, Congress is not in session to debate or take up any proposals and even if details were to miraculously appear and Congress came back to pass the President's plan, the final problem is that it is very unlikely it would pass both chambers of Congress," Pomerleau said.

Larry Kudlow, the top economic adviser to the President, said Tuesday that Trump follows through on his promises, but Kudlow also acknowledged that the proposed tax cuts may not materialize in the near term, saying, "It may not surface for a while ... but that's his goal."

Given the heavy lift and tricky political waters the House GOP leadership team will be facing as it attempts to prevent a government shutdown by the December 7 funding deadline -- one the President has all but called for if he doesn't secure significant financing for his border wall proposal -- there's limited appetite to move forward on anything more substantive, aides said, even as they left the door open to something happening if the President's economics team sent something substantive to the Hill.

Prior to the introduction of Trump's new tax cut idea, Senate Majority Leader Mitch McConnell of Kentucky made clear that his priorities for the lame-duck session of Congress are funding the government and confirming more conservative judges for district and appellate courts -- something Senate Republicans have done at a record-setting pace throughout the 115th Congress.

While House officials were more careful about addressing the future of the President's nonexistent tax cut proposal, colleagues in the Senate were less circumspect.

"It's a talking point," was how one senior GOP Senate aide put it. "It's something to talk about in the lead-up to the election. Nothing more."

A congressional leadership aide similarly told CNN that Congress controls the levers of taxation power "but it gives the President something to talk about" on the campaign trail.

That aide added, "To Trump's credit, he has the ability to will things into existence from nowhere."