Ryan told his rank-and-file members in a closed-door conference meeting that when the votes are there, he will put the bill on the floor, in an implicit admission that the majority of 216 votes that he needs to ensure passage is not yet in place, one member inside the meeting told CNN.
President Donald Trump, at an event for the Air Force Academy football team, said Tuesday "I think it's time now" for a vote on the bill.
But at least 22 Republicans plan to oppose the current bill, according to CNN's ongoing whip count. Influential Rep. Fred Upton of Michigan, a former chairman of the House panel that oversees health care issues, said Tuesday he is against the measure.
Sources said House leaders are working with members one-by-one to get the votes they need to get to 216. Health and Human Services Secretary Tom Price is going through the fine print of the latest version of the bill with undecided members to try to assuage their concerns on specific items.
"The President can't do this," a Republican close to the talks said. "This is fine print time."
As for timing, a senior Trump administration source told CNN's Dana Bash that House leadership was looking at the possibility of a vote Thursday and that they were within five votes of the magic number of 216.
Others are not going that far.
"We will have a vote when we have the votes," a senior GOP leadership source said.
A member who is leaning "yes" on the revised bill said he thought the chances of a vote this week were "50-50."
No changes to the bill -- yet
"Speaker has to get to 216," the member said. "He's working it one by one."
Roughly 16 members are undecided, but thus far, there are no public tweaks to the bill. Speculation has centered around additional money for high risk pools, for instance.
"Obviously some members are looking for changes, but we've not made any at this point, and don't know that we will," a GOP leadership aide said Tuesday afternoon.
As originally introduced, the bill would leave 24 million fewer people insured by 2026 than under Obamacare, the non-partisan Congressional Budget Office said.
The bill would make major changes to the Affordable Care Act, eliminating the taxes on wealthy individuals, industries and others, as well as jettisoning the individual mandate that requires Americans to purchase insurance or face a penalty. It would also replace Obamacare's subsidies, which are based on income and cost of coverage, with refundable tax credits based mainly on age.
It would allow states to relax some key Obamacare protections of those with pre-existing conditions. States could apply for waivers to allow insurers to offer policies with skimpier benefits and to charge more to those with medical issues if they let their coverage lapse.
The speaker and House Majority leader Kevin McCarthy used the conference meeting to stress the urgency and the stakes of the push to repeal Obamacare, sources in the room said.
But in a subsequent news conference, Ryan conspicuously did not say that he had the votes to pass the measure or that there would be a vote this week, despite confident predictions by top White House officials on Monday that passage was close.
A key GOP source close to the health care process sounded gloomy about the prospects for the repeal and replace bill this week in the House.
"There are enough people to prevent us from having the votes," the source said.
"Not sure they'll be there or ever be there," the source said of the votes in favor of the bill, adding that possible likely the House will "keep kicking the can" from one week to the next.
As the effort to pass the bill reached a crucial point, some members appeared increasingly exhausted by the repeated push on health care -- a factor that could see some shelve objections just to offload it to the Senate.
Given the political consequences of failing to pass Obamcare repeal after weeks of effort, some members appear to believe that the vote will succeed in the coming days. If Ryan cannot schedule a vote, Republicans could face a backlash from their base when they return home for next week's recess following years of promises from the GOP to repeal and replace Obamacare.
But Rep. Tom MacArthur, R-New Jersey, the author of the latest compromise designed to finally get the bill to a vote, told reporters Tuesday morning: "I think it's close."
House Majority Leader McCarthy, R-California, agreed.
"Very close!" he said.
McCarthy said Rep. Lou Barletta, R-Pennsylvania, is now a "yes." Barletta was previously undecided because he had concerns that people would defraud the system and apply for tax credits without the government checking Social Security numbers. Now he's been promised that there will be a separate bill to fix that issue later.
Trump has made several interventions, but it was unclear whether his assurances reflected in-depth knowledge of what the American Health Care Act contains, and whether he was helping or hindering the push to pass it.
Ryan however said the President had been "nothing but helpful" on health care.
"We are making very good progress with our members and our president has been instrumental in that," he said.
Fight over pre-existing conditions
A vital sticking point in the bill is the issue of protections from pre-existing conditions.
"Pre-existing conditions are in the bill. And I mandate it. I said, 'Has to be,'" Trump said on CBS's "Face the Nation" Sunday.
The President added that that "we actually have a clause that guarantees" coverage for those with pre-existing conditions."
In the latest version of the bill however, insurers could charge people with pre-existing conditions like cancer and diabetes higher rates than other healthier patients if they allow their continuous coverage to lapse.
The measure would allow states to opt out of a mandate contained in the Affordable Care Act that requires insurers to charge the same rates for patients with pre-existing conditions as for healthy people.
The Republicans would have states set up "high-risk pools" for some of those with pre-existing conditions -- but some observers fear that the cost of such coverage could be hugely expensive.
Such a scenario is one reason why moderates are having trouble lining up behind the bill.
But one previously undecided member, Rep. Jim Renacci, who is also running for governor of Ohio, said he would vote yes on the bill because he was satisfied that states would be answerable to their people on protections for pre-existing conditions.
"If the governors decide to remove that, that's something they are going to have to do within the constraints of the people they represent," Renacci told CNN's "New Day" on Tuesday.
What's in the bill?
The GOP health care bill would eliminate Obamacare taxes on the wealthy, insurers and others, and get rid of the individual mandate imposed by Obamacare, officially known as the Affordable Care Act. Instead of the Obamacare subsidies that are tied to income and premiums, the GOP plan would provide Americans with refundable tax credits based mainly on age to purchase health insurance.
The legislation would also allow insurers to charge higher premiums to those in their 50s and early 60s, compared to younger consumers.
The bill would also significantly curtail federal support for Medicaid and allow states to require able-bodied adults to work. After 2020, states that expanded Medicaid would no longer receive enhanced federal funding to cover low-income adults, and those that hadn't expanded would be immediately barred from doing so.
And it would allow states to relax some key Obamacare protections of those with pre-existing conditions, which are among the health reform law's most popular provisions. States could apply for waivers to allow insurers to offer skimpier policies that don't cover the 10 essential health benefits mandated by Obamacare. Also, insurers would be able to charge higher premiums to those with medical issues if they let their coverage lapse. States requesting waivers would have to set up programs -- such as high-risk pools -- to protect insurers from high-cost patients.
However, the GOP bill doesn't touch one another beloved piece of Obamacare -- letting children stay on their parents' insurance plans until the age of 26.