Plans have been drawn up to expand the Hinchingbrooke Country Park car park by 250 spaces to alleviate parking pressures on Hinchingbrooke Hospital.
The proposal is part of Huntingdonshire District Council's business case to take a longer lease for the site in Huntingdon from the county council, which owns the freehold.
The district council, which currently subsidises the country park at an annual loss of £70,000, said it wants to invest in the facilities, but first wants confirmation of continued control over the longer term.
The county council has already rejected the idea of handing over the freehold, but will now consider a 99-year lease.
But the county council will also consider the option of taking back ownership of the site, and has agreed to open up talks to explore "a more commercial focus".
The decision to look at alternative operators to the district council was made at the county council's commercial and investment committee on May 24. A successful opposition amendment means alternative commercial opportunities will now be carried out "in collaboration with" the district council.
The committee chairman, Councillor Josh Schumann, outlined Huntingdonshire District Council's proposal for £1.5million worth of investment, which would extend the car park and improve the cafe and leisure facilities, but said he thought it could be "more ambitious".
Fellow Conservative and vice-chairman of the committee, Councillor Anne Hay said she was "thoroughly underwhelmed" by the district council's business plan, and added "there is so much potential there they have just not even looked at or it would appear they have not looked at".
The county council's deputy chief executive and chief finance officer, Chris Maylon, said: "There is a lot of tension between the district council and the county council" over the issue.
And he added dialogue between the two had, at times, been "quite confrontational".
The district council wants to invest to increase the area's potential to make money, and so reduce losses. The lease arrangement with the county council expires in April 2026, but the district council wants it extended before then so that it can benefit from any investment and secure the site's future.
Mr Maylon said the district authority had sent the county council an e-mail saying: "if you don't agree with our proposal you can take back the keys".
Liberal Democrat, Councillor David Jenkins said following the meeting: "I was staggered at the apparent animosity between the two councils. They are both Conservative councils after all."
Following the committee's decision, the county council will now enter talks to examine the possibility of a private operator taking over.
But Cllr Schumann said after the meeting that whatever the outcome, additional parking is likely.
He said: "I would envisage that the parking is always going to be the profitable area of the delivery of the country park.
"I think it is fairly likely that any forthcoming offer is going to be underpinned by an increase in parking provision."
The country park's car park is within walking distance of the hospital, and it was suggested in the meeting that more hospital staff and visitors paying to park there could help finance the country park.
Cllr Schumann gave Milton Country Park as an example of one way a private operator could be involved.