Republicans are also waiting to see whether Trump will lay out more details about his objectives, particularly after the incoming president warned party leaders to be careful how they proceed with repeal.
"We're continuing to formulate a plan," Sean Spicer, a transition spokesman, told reporters Tuesday in a conference call.
On the House side, the new chairman of the conservative Freedom Caucus said his group wants to see more details about an Obamacare replacement before voting on the budget resolution.
"We hope they would see the prudence of waiting," Rep. Mark Meadows, R-N.C., said Monday night.
Interviews with many Republicans indicate that the party is no closer to consensus two months after an election that gave them unified control of the White House and Congress.
Even before the new amendment was offered, Cassidy, Collins and other senators pushed to delay any repeal of Obamacare tax hikes so that there would be revenue to pay for a replacement plan. This puts them at odds with House conservatives, who have been demanding a full, immediate repeal.
On Monday, more senators said they agreed with a delay on the tax front, including Sen. Mike Rounds of South Dakota.
John Cornyn of Texas, the No. 2 Senate Republican, told reporters that the taxes used to subsidize insurance for millions of Americans could be dealt with later this year in a larger tax overhaul.
Republicans senators are also grappling with the risks of repealing the law before a replacement is ready.
Sen. Ron Johnson of Wisconsin said he wants to see a "pretty darn specific" proposal to replace Obamacare before it's repealed.