The process has invited ire from The New York Times editorial board, which wrote on Tuesday that Republican senators were working on a bill that could take away insurance from millions of people - all while refusing to subject the bill to public scrutiny.
Capitol Hill sources who asked not to be named said Trump described the version of the legislation that passed the House last month as "mean".
Conservatives are also frustrated that Senate leaders are mulling plans to make tax credits for buying insurance more generous for low-income people.
Facing expected unanimous Democratic opposition, Republicans will be unable to pass a Senate bill if just three of the 52 GOP senators vote "no".
"The total bill hasn't been resolved", he said. "We can't afford to fail; we've got to get it done, and he's certainly fine with the Senate taking a different direction than what the House [did]".
A slower roll back of Obamacare's taxes would generate enough revenue for a more generous health care bill, although retaining Obamacare's one trillion dollars in taxes would generate significant backlash from conservatives.
"Americans won't forget that @HouseGOP passed a "mean" bill to rip healthcare from millions then celebrated @ the WH", said Rep. Elijah Cummings, D-Md. "We had some of our members that brought that up".
It didn't help the optics after the fact when some Republican lawmakers were forced to acknowledge that they had not read the bill before voting in support of it. They agreed to language letting states drop requirements under Obama's health care law protecting people with pre-existing medical conditions from higher premiums and requiring insurers to cover specific services like maternity care. Considering a number of offsets and spending, over the next decade the bill would save the government a total of $119 billion.
Americans deeply disapprove of the AHCA.
White House spokeswoman Sarah Huckabee Sanders declined to comment, telling reporters aboard Air Force One Tuesday evening, "We don't comment on rumors or private conversations". "How do we do it?'"
Negotiations this week have centered on the Affordable Care Act's individual marketplaces, including a proposal that would allocate billions in funding to smooth over the transition to a new model should the ACA repeal succeed, The Wall Street Journal reported (sub. req.). The 2010 ACA bill known as Obamacare sought to get all Americans on some form of health insurance.
Senate Republican leaders say they still hope to vote on a bill to repeal Obamacare before lawmakers leave town for the July 4th recess. McCaskill went on to say that she has "no idea of what's being proposed".