One adviser to a senator present at the meeting said the acknowledgment was simply that "they don't have the votes to pass this in current form," a reference to whether the House GOP bill in its current form could survive in the Senate.
The meeting included a handful of conservative senators who have expressed deep concerns with the bill. According to multiple aides, Vice President Mike Pence made a short visit to the closed-door briefing with lawmakers.
Another senior Republican aide to a senator in the room said that the White House is "definitely looking at pretty big changes to the Ryan bill."
"There is no possible way a bill can be drafted by an individual or a handful of people and shoved down the throats of 535 people. It doesn't work that way," a Republican senator told CNN, noting that changing the bill is a normal part of the process.
One of the senators who attended, Ted Cruz of Texas, said talks are "productive."
"The conversations continue to be productive and they're moving forward well. The conversations are ongoing with the White House, the administration and the House and the Senate," Cruz said Wednesday.
"The House has a current bill that personally I don't think gets the job done. I think we need more work on the House bill," Cruz said later at the US Hispanic Chamber of Commerce legislative summit.
A senior administration official said in the meeting of the claim the House bill cannot pass the Senate as currently written said, "I think they made that claim. We didn't rebut it."
This official confirmed that the White House has continued to advocate for changes to the bill that will help secure conservative votes in the House -- but also some tailored to moderates' concerns.
"I think we made a lot of progress in changing language that's going to be appealing," the official said.
The official declined to elaborate on which changes have been made already, but confirmed that the White House's position privately is still to call for an earlier sunset of the Medicaid expansion.
House bill in peril
The House legislation has run into trouble in recent days. After the non-partisan Congressional Budget Office warned Monday that 24 million Americans could lose insurance coverage over the next decade if the House bill was passed, moderate Republicans Reps. Leonard Lance and Rep. Ileana Ros-Lehtinen have come out publicly opposed to the legislation and others have been vocal about their concerns.
Moderates are only a piece of the puzzle, however. House conservatives -- especially members of the Freedom Caucus -- have been vocal in their opposition to the bill in its current form, and House Speaker Paul Ryan can only afford to lose 21 of his own members on the bill.
If House leaders address conservatives' concerns by rolling back Medicaid expansion sooner than 2020, they risk losing moderates. If they try to win over moderates by making tax credits more generous for older Americans, they could lose more conservatives.
Emerging as another top concern from the White House is the question of whether the House bill can even make it through the Senate under reconciliation rules or the so-called Byrd Rule. Lawmakers have long agreed that the best way to repeal and partially replace Obamacare is through the process of budget reconciliation, which only requires a simple majority vote in the Senate. However, budget reconciliation comes with a whole host of requirements. Members cannot just pass anything through reconciliation.
According to one senior Republican aide to a senator in the room Tuesday, there is growing recognition that some aspects of the House bill may not even be fully accepted under those rules. In that case, the Senate parliamentarian could rule that pieces of the legislation aren't allowed under reconciliation.
Some conservatives are arguing to the White House that the Senate should override the parliamentarian, but if they are going to take that political risk, they should do it with a bill they believe will lower premiums in upcoming years, not the Ryan bill.
The Republican aide to a senator in the meeting said some senators are pushing for a more robust package that would allow states to sell insurance across state lines, make additional changes to Medicaid and help lower drug prices instead of overriding the parliamentarian to pass through the House package.