Confusion reigns over the future of the unpopular GP co-payment amid suggestions it will be shelved or that the government could seek to bypass a hostile Senate to introduce some version of the policy through regulation.

The Australian Medical Association (AMA) has called on the government to declare its position, noting that Health Minister Peter Dutton had this week ruled out the possibility the government would bypass the Senate because it had legal advice indicating it could not introduce the fee without legislation.

But on Thursday morning, Mr Dutton told reporters in Canberra he would not "rule things in or out" and the government was "determined to make Medicare sustainable".

In an interview with Radio National on Thursday morning, the government's leader in the Senate Eric Abetz also refused to speculate "as to different methodologies that might be employed" to introduce some form of GP fee.

Fairfax Media and other outlets reported on Thursday that Prime Minister Tony Abbott has privately conceded defeat on the unpopular fee proposal and will shelve the policy before the end of the year.

AMA president Brian Owler said "it would nice to have certainty from this government about what they're doing".

"Clearly there are a number of messages coming from the Prime Minister's office but then we've got Eric Abetz out there this morning defending the policy," Dr Owler said.

"It would be nice to just have a clear announcement from the government and then we could move on and have a more complex and nuanced discussion about health policy."

Dr Owler said he did not believe the government would attempt to bypass the Senate.

"They've already looked at it – you can't introduce a co-payment without legislation, you can't change the [Medicare] rebate without legislation," he said.

"It would be a very surprising and cynical and poor political move to try to do that."

Medical research groups urged the government on Thursday to find a way to establish its $20 billion medical research future fund, with or without a GP co-payment.

Revenue from the $7 fee was to be used to establish the new fund for medical research.

Professor Brendan Crabb, director of the Burnet Institute and immediate past president of the Association of Australian Medical Research Institutes, said there was universal support for the medical research fund, even among opponents of the GP fee.

"There simply must be a way to get this $20 billion nation-shaping fund up," he said.

"It would be an absolute tragedy if this decision came through and led to that fund not happening.

"We implore the government to find a way to make it happen."

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