But the problems that plagued the GOP two weeks ago -- divisions among moderates and conservatives -- haven't gone away.
After a short conference meeting Tuesday morning, House Speaker Paul Ryan told reporters what he'd already told his fellow Republicans: there is no new agreement or bill, only ongoing and positive conversations.
"We're throwing around concepts to improve the bill," Ryan stressed. "That's occurring right now, but that is not to say that we are ready to go. Because we want to make sure that when we go, we have the votes to pass this bill."
He added, "I don't want to put some kind of an artificial deadline because we're at that conceptual stage."
Movement on health care accelerated Monday when Pence met separately
with both sympathetic members of the moderate Tuesday Group and the entire House Freedom Caucus
Tuesday night, Pence will meet with more lawmakers, said Rep. Tom MacArthur, a co-chairman of the moderate Tuesday Group.
"Some of us will be meeting with him tonight," said MacArthur, a New Jersey Republican. "It will be a little broader group. We went there last night and he'll meet with a broader group tonight to try and flesh this out."
One Senior House GOP aide said to expect "a number of additional meetings" with various groups and members, but cautioned that the latest round of meetings still doesn't mean leadership plans to move ahead with a vote soon.
Ryan, Pence and Trump budget director Mick Mulvaney met Tuesday afternoon, a senior leadership aide said. The aide downplayed the meeting, however, saying Ryan and Pence regularly touch base when the vice president is in the Capitol.
On the other side of the Capitol, the second ranking Republican in the Senate, John Cornyn, downplayed chances of a revived health care plan.
"God bless them," the Texas Republican said. "I think they've got a way to go. I heard the speaker talk about 'conceptual.' They were at concepts. So, they have a way to go but if they send us a bill, we'll take it up."
Let states opt-out from Essential Health Benefits requirement?
The proposal being floated by the White House at this point would give states the opportunity to opt out of some Obamacare regulations including the requirement that insurers cover Essential Health Benefits
like maternity care or substance abuse, a version of which was already included in the previous House bill. Members of the Freedom Caucus are also looking for the White House to give states the ability to opt out of the community ratings provision, which currently protect consumers from insurers charging higher premiums to individuals based on their their medical history or gender.
Sen. Rand Paul, whose interests on the issue have been closely aligned with the House Freedom Caucus, called the suggestion that states can opt-out of the community rating provision, "a step in the right direction."
The Kentucky Republican, who discussed health care with the President while golfing over the weekend, said he, like the Freedom Caucus, still needs to see this in writing.
But GOP leadership aides still insist that talks of reviving the bill for a vote on the floor this week are premature and there are some fears that the proposed change floated by the White House could trigger backlash from the moderate wing of the party that was already hanging on by a thread. An aide emphasized that changes had to "add net votes" to the total in order for leaders to agree to move forward with it.
During the closed-door conference meeting Tuesday morning, some members suggested that the House needed to act soon on health care and not leave for recess without action, but that was not a widespread sentiment, and other House Republicans at the closed door meeting suggested there was a "need to do this right."
Rep. Steve Womack, a Republican from Arkansas, said leaders told members on Tuesday that they "continue to discuss" options but that there was "no white smoke coming out of any leadership office that I'm seeing."
"We can't try again and fail," Womack said.
Enough to attract moderates?
Multiple House GOP members told CNN they still don't see how the latest floated compromise changes the math to get to 216.
Mike Coffman, a Republican from Colorado who hails from a swing district and had already pledged his support to House leadership's previous health care bill, expressed concern about removing the community ratings provision.
"I don't see how that would work," Coffman said adding that he still needed to see "what else is in the bill."
And, there was still little evidence that Pence's renewed push had changed any minds in the House Freedom Caucus.
Walking into the GOP conference meeting Tuesday, Rep. Justin Amash, a Freedom Caucus member from Michigan, repeated that no deal had been reached.
And Rep. Mo Brooks, another House Freedom Caucus member from Alabama, said he still thought what the Vice President offered last night was still "inadequate to change the two fundamental problems with Obamacare." Brooks said he still thought the best solution would be to bring up the 2015 repeal bill that the House and Senate already voted on.
"I am baffled that we are not passing that same legislation today that we know we have the votes to pass because we just did it a year and a half ago," Brooks said.
Aware of the divisions within his own party, Ryan cautioned reporters Tuesday about the state of Republicans' progress on the issue, saying what's important is that members are talking through their differences.
"We have very productive conversations occurring among our members, but those are productive conversations," Ryan said. "That doesn't mean that we have language and text that's ready to go and the votes are lined up. But that's what you need to do in order to get that, meaning get members talking. That's happening."