(CNN) Republican negotiations on a House immigration bill that would fix the Deferred Action for Childhood Arrivals program are in the final stages, key lawmakers said as they left a secretive meeting in the House basement on Wednesday.

Both moderates and conservatives are coming together on an outline of a bill brought on by weeks of negotiations behind closed doors, as leadership brought the two wings of the party together to avert rebellions on both sides. Some of the Republican immigration negotiators are also touting White House involvement in trying to unify support behind a plan.

According to a source in the room, House Speaker Paul Ryan said at Wednesday's meeting with his conference that he has spoken to President Donald Trump about the compromise bill coming together and that Trump is "excited" about the deal being put into text. Separately, Rep. Jeff Denham, a moderate Republican from California involved with negotiations, said Ryan had spoken with the President and he had been supportive.

After a breakthrough agreement on how to proceed Tuesday -- and arm twisting by leadership -- that cut off moderates' efforts to buck leadership control of the floor, talks Wednesday centered around hammering out the details of the policy itself.

The progress in negotiations sets the stage for votes on immigration on the House floor next week, which will include a vote on a conservative proposal that is not believed to have the support to pass and a separate compromise being written that will stem from the negotiations currently in progress.

Though the bills' fates are still unclear and it's possible neither passes the House -- let alone moves in the Senate -- the prospect of Republicans having a debate and vote on the political third rail of immigration on the House floor the summer before midterm elections was unthinkable just months ago.

"We're just doing the cleanup stuff from the negotiations that (Reps) Raul (Labrador) and Carlos (Curbelo) did yesterday," said conservative Freedom Caucus Chairman Mark Meadows as he left member negotiations Wednesday. "So we're just trying to dot our I's and cross our T's."

"We're just about there," Curbelo said. "I think we'll definitely see text this week."

Early support from the White House

In addition to the speaker relaying the President's support, Trump aide Stephen Miller, a hardliner on immigration policy, stopped by the conservative Republican Study Committee's lunch Wednesday on Capitol Hill to try and build support for the immigration compromise that is being put together.

His message, according to a person in the room, was that the administration supports both the conservative bill lead authored by Rep. Bob Goodlatte -- which is the more conservative option but is not expected to have enough votes to pass -- but is also open to the the compromise bill that is taking shape right now.

The public show of support from the White House is a substantial boost for the bill, as lawmakers have long said they could not support anything the President would not sign. Thus far, the White House has largely stayed away from the House negotiations, at least beyond offering input to the close group negotiating.

But an overt sign of support from the President could be enough to push an immigration bill that would otherwise falter through Congress.

Whether it would be enough to get it through the Senate is up to Senate Majority Leader Mitch McConnell, who already went through an unwieldy and unsuccessful immigration debate in his chamber earlier this year and has thus far shown no interest in tackling the issue again.

Conservatives were quick to pour cold water on the notion that the White House is fully behind the compromise.

"I think at this point there are concepts the President supports, I think that's what the speaker has said," Meadows said. "But at this point, no one in the White House has seen any legislative text. No one on Capitol Hill has seen any legislative text. And so to suggest that they're strongly supportive of this is just inaccurate."

In response to Miller -- a known conservative hardliner on immigration -- vouching for it, Meadows shot back:

"Ask Stephen if he's seen a bill. Maybe he saw one in North Korea, but I didn't see one in Washington DC."

What's in it

CNN has obtained a draft, from a source close to the negotiations, of the outline lawmakers are working from to write the bill, which, when described to Curbelo, was confirmed as largely still what they're working on minus a few "details filled in." The broader GOP conference was briefed on the toplines of the bill in a Wednesday morning meeting.

The bill is designed to hit the conceptual "four pillars" that Trump has asked for in an immigration deal: a solution for DACA recipients, changes for border security, cutting family-based migration and ending the diversity visa lottery.

On the DACA piece, to protect the young undocumented immigrants who came to the US as children that were covered by DACA, which Trump has decided to end, the bill would create a new visa that would include DACA-eligible immigrants but also be broader, so conservatives are satisfied it's not a "special pathway" to citizenship for undocumented immigrants alone.

The proposal would allow both individuals who qualify for the new iteration of a DACA permit and children of legal immigrants who have grown up in the country but may be aging out of status to apply for a green card after a waiting period.

To make room for those new visas, other parts of legal immigration would be cut accordingly, on the family side and diversity visa. The plan would cut visa categories for married adult children of US citizens and adult siblings of US citizens.

The cut visas from family categories and diversity would be reallocated to the new green card category and employment categories.

It would appropriate $25 billion for the wall and border security.

It would also include some of the administration's requests for enforcement powers, including making it harder to pursue asylum claims in the US and making it possible to immediately return unaccompanied children to Central America unless they establish they have a claim to asylum under the new higher standard.

It would also change a court settlement governing the treatment of children in government custody, including "ensuring alien minors apprehended at the border along with their parents are not separated from their parent and legal guardian."

The details on how the settlement would otherwise change is still being drafted, Curbelo said when asked about that provision.

Curbelo said he believes the bill is picking up support.

"The reports and feedback I'm getting back from conservatives is that there's growing momentum," Curbelo said.

Conservatives, meanwhile said they needed to see bill text.

"I think that's still not a given," Freedom Caucus leader Jim Jordan said when asked if conservatives could back what's been negotiated. "But i think members are open."

This story has been updated.