A Defence spokesman said Bupa Health Services had been "ranked as the preferred respondent to enter into negotiations".
"Medibank Health Solutions, the incumbent, while considered suitable, was ranked second," the spokesman told AAP on Wednesday.
"Respondents have been provided verbal advice of the outcome of their submission. The procurement process is ongoing."
While not mentioning Bupa, AMA President Dr Tony Bartone said in a statement on Wednesday the government and successful tenderer needed to work closely with doctors to ensure ADF personnel received the best possible care under the new arrangements.
"Our defence personnel deserve quality health care, but we have witnessed problems in the past," he said.
"The AMA was highly critical of the Medibank Health Solutions approach to the provision of medical services when it was first awarded the contract in 2012.
"Many GPs and other medical specialists refused to provide care under the arrangements implemented by MHS.
"There was widespread concern about MHS decisions, including significant fee cuts, unfair contracts, its control of referral arrangements, and poor consultation with the profession - particularly in the initial period after the contract was awarded."
He said the new provider needed to "avoid making the same mistakes that disadvantage defence personnel and doctors".
Medibank chief executive Craig Drummond said on Monday written confirmation was being sought, having only been verbally advised of the decision.
"We are disappointed, unquestionably."
Medibank predicts exit costs from the contract of about $5 million to be incurred in the second half of the 2019 financial year.
Medibank last year made an operating profit of about $30 million from the ADF contract, which it has held for six years.
It said a high level of care had been provided to more than 60,000 permanent and 20,000 reservist ADF personnel, consistently meeting key performance indicators.
Services included on-base health support, pathology, imaging and radiology and a 24-hour ADF national health hotline.