But the reality is that not everyone can get what they want from House Speaker Paul Ryan, who is trying to get the 216 Republicans he needs to vote for the bill without any Democratic support. He can afford to lose 21 members; 20 are on the record leaning against it, according to CNN's latest count.
And as leaders listen to complains, Ryan and others are already counting votes. The speaker was spotted having a long conversation on the House floor Thursday afternoon with Rep. Gary Palmer, an Alabama Republican opposed to the current bill.
One change that House leaders are considering is adding a work requirement for able-bodied adults who receive Medicaid. The change may appease some conservatives without alienating moderates that leadership needs to hold on to.
One Republican member with direct knowledge of the negotiations said that there is no "arm twisting " at this point and no decisions have been made yet on how to amend the bill to get it passed on the House floor.
"Anytime you have a large bill like this, it's a balancing act. I think all sides are going to try to get their issues addressed. I don't think it's leaning one way or another," the Republican said.
The member cited specific concerns they have heard in whip meetings including moving up the Medicaid expansion expiration date, making tax credits more generous based on income level for older Americans as well as adding a work requirement.
But the member noted that there are some members who may never get to yes and whips are taking that into consideration.
"There are always going to be people who serve under the dome ... that any government injection into the health care system is too generous," the Republican said.
After the legislation advanced out of the Budget Committee
Thursday morning, Republicans' best shot for changes to the bill will come next week in what is known as a "manager's amendment," which incorporates the tweaks that leaders believe will help smooth passage. That is attached to the legislation in the Rules Committee, the final panel that reviews the bill before a floor vote.
House leaders trying to hold the line say fellow Republicans should make their concerns heard but not try to derail the bill simply because it doesn't include everything they want.
Oklahoma Republican Rep. Tom Cole, an ally of the leadership, urged those who are vowing to oppose the bill because it doesn't contain all the elements they want to keep the bill moving instead of threatening to derail it.
"None of these votes to me are votes that you ought to live or die by -- they are process votes to move the product along," said said. "You ought to try to continue to amend it, to change it."
'It's a balancing act'
Florida Rep. Mario Diaz-Balart told CNN he still had "concerns in a number of different areas" with the leadership's bill but said he believed the leadership had set up a process to hear out members who want changes.
Pressed how he would vote on the bill, Diaz-Balart declined, saying, "it's too early to tell if it will be a bill that I can whole heartedly support or not. Right now it's a work in progress."
Both Diaz-Balart and Cole sit on the Budget Committee and voted Thursday to advance the bill. Palmer was one of three Republicans on the panel to vote against it.
Ryan told reporters that President Donald Trump is personally involved in lobbying House Republicans, and multiple GOP aides say they believe the President could be the best advocate they have as they close in on a final vote.
"He is a constructive force to help us get to a resolution so that we get consensus on how to repeal and replace Obamacare," Ryan said Thursday.
House vote next week?
Cole told reporters he expected a Rules Committee markup on Wednesday, but GOP leadership aides have said that while the goal is still to vote next week, efforts to tweak the legislation could shift that timeline.
Members continue to aggressively lobby to make their changes.
Mark Meadows, the Chairman of the conservative House Freedom Caucus, told reporters Thursday he was working with his members to introduce an amendment package as soon as Friday he hoped would make the current GOP health care bill to repeal and replace Obamacare palpable to his members.
Meadows wouldn't elaborate about what he wants to see in his package, but members of the Freedom Caucus have said in the past that they have concerns about the legislation's tax credits to help people buy insurance as well as the fact that the House bill does not sunset Medicaid expansion until 2020.
Even if 'St. Peter called ...'
But for every member of the conservative wing of the party that leaders try to appease, they risk losing a moderate Republican.
Rep. Charlie Dent, a moderate from Pennsylvania, has said repeatedly that cutting off the ability of people to enroll in the Medicaid expansion before 2020 is a non-starter for him. Also weighing heavily on moderates like Dent is the fact that they may be forced to day a controversial vote in the House that won't have a chance of passing the Senate.
"If I hear the Senate say one more time, that this bill's dead on arrival, you know my head will explode."
In an attempt to give their members more cover, leaders plan to move two stand-alone health care bills to the floor next week before the vote on the repeal package. One would allow insurers to sell insurance across state lines and one allows small businesses to pool coverage plans. Both are expected to pass the House but will need 60 votes in the Senate, a bar that even conservatives who support these proposals say is unlikely.
North Carolina GOP Rep. Walter Jones said the only thing leaders could do at this point to satisfy him would be "to pull the bill and say we are going to do something entirely different than what the Democrats did."
He said leaders needed to slow down and told reporters he felt zero pressure from supporters of Trump to get behind the bill.
"If St. Peter called and said you need to vote for this bill if you want to get into heaven, I'd say I can't do it," Jones said.