It's understood Prime Minister Tony Abbott is preparing to downgrade or axe the proposal, which would charge patients $7 on bulk-billed GP visits.
The measure could be ditched as part of a “barnacle” clearing exercise as early as Thursday, but such reports have been dismissed by Health Minister Peter Dutton and Government Leader in the Senate Eric Abetz.
Speaking to media on Thursday, Mr Dutton said Australia needed to "move away from the view that everything is free".
“Medicare is dead if we can't make it sustainable," he said.
"These are plans which will set up Medicare for a decade or two.”
The co-payment, which was unveiled in the May budget to fund medical research, is unlikely to pass due to a lack of support in the Senate.
The government is engaged in last-minute lobbying to save some of its key budget measures before rising for the year on December 4.
Mr Dutton said negotiations were continuing with senators, but he said the Labor Party needed to be "more repsonsible".
“The Labor Party will kill Medicare by offering everything to everybody for free," he said.
Senator Abetz has also stood by the proposal.
Speaking to ABC Radio, he said the policy remained part of the government’s legislative agenda.
“It is difficult? Of course it is,” he said.
“I’m not going to say that anything is possible or impossible. It is good policy, it is our policy.”
Senator Abetz said the measure had previously been championed by Labor politicians such as former prime minister Bob and current Assistant Treasurer Andrew Leigh.
He said it was now up to Labor to support what they knew to be “right policy”.
“Those in the policy space know it’s good policy,” he said.
