Washington(CNN) House Speaker Paul Ryan attempted to walk a fine line to keep the health care debate moving forward without putting House GOP members in a box on how it will deal with a scaled down version of the Obamacare repeal-and-replace bill that could emerge overnight from the Senate.

Pressed by a group of Senate GOP members with an unusual request of guaranteeing the House won't vote on the bill they could pass in the early morning hours, Ryan pledged if that's what it takes, he was on board.

"Senators have made clear that this is an effort to keep the process alive, not to make law. If moving forward requires a conference committee, that is something the House is willing to do," the speaker said in a written statement Thursday night.

But Ryan had a demand of his own, insisting the Senate had to come up with something it could get through, saying, "we expect the Senate to act first on whatever the conference committee produces." And he didn't specifically state what some Senate Republicans wanted him to guarantee -- that the House would never take up just the "skinny repeal."

His comments followed several Republican senators who said earlier Thursday that they wouldn't vote for a so-called "skinny" health care bill unless if went to conference committee for changes. That plan appeared to have the support to pass the Senate. House GOP leaders already put members on notice they may be working this weekend if it gets dropped in their lap and they scheduled a meeting for all House Republicans for Friday morning.

Already many House conservatives are signaling they don't want to just vote on the stripped down Senate Republican bill.

"We don't want a skinny deal to be some kind of bony skin," Rep. Mark Walker, Chairman of the Republican Study Committee, told reporters Thursday, saying he prefers hashing out a broader Obamacare repeal and replace package in a formal negotiation with members of both the House and Senate.

House Freedom Caucus Chairman Mark Meadows, a conservative leader who helped shape the House bill that passed in May, made it clear he wasn't a fan of the latest Senate bill. He told CNN if Senate Republicans can muster the votes to pass it, "the only response if it passes would be to go to conference. We can't send that to the President. I mean, after seven years and seven months, that's the best we can do?" Meadows also declared that the skinny bill didn't have the votes to pass the House if leaders tried to put it on the floor.

On Thursday a number of Senate Republicans said the leadership strategy now behind moving ahead with the skinny proposal was mainly to move onto another stage of discussing a final bill with their House counterparts. In an unusual twist, many senators were openly saying they didn't want the House to pass their bill, but instead they just viewed the measure as a vehicle to punt on finalizing the details later.

House Majority Leader Kevin McCarthy sent out a notice to all House members instructing them: "All members should remain flexible in their travel plans over the next few days."

Rep. Bradley Byrne, R-Alabama, also criticized the scaled-down Senate bill, telling CNN the strategy he favors is putting leaders of both chambers together and having them decide what framework for a health care overhaul can actually make it through both the House and Senate. He thought House members could be voting on a procedural measure - a "motion to instruct" - on Saturday to set that in motion.

Many House conservatives indicated throughout the health care debate that their test for supporting any proposal was if it brought down health insurance premiums, and the analysis on the latest version of the Senate bill would be a significant increase in those costs."

"I don't think it helps with the insurance problems that we've seen under the present law. I don't think we can responsibly just let the insurance market continue to deteriorate. We need to do something the sooner the better," Byrne said.

Walker echoed the same worry, "the premium spike would be a concern for us."

There is still confusion among House Republicans about the contours of the Senate GOP leaders' plans, but chatter has intensified among rank-and-file members that they need to pay attention now because there is a growing belief that after watching the Senate struggle through the process the ball would soon be in their court.

"Every time I talk to a different senator I get a different understanding and so that leads me to believe that they are still figuring out how to get there," Byrne said.