Durham are in talks with the England and Wales Cricket Board (ECB) following claims they could face a points deduction for receiving financial support from the governing body.
Advanced payments were made to Durham by the ECB in May 2016 to ensure the club could carry on playing.
The monies related to fees for staging the Sri Lanka Test later that month.
"Talks continue with the ECB in relation to securing a long-term future," a Durham statement read.
The county finished fourth in County Championship Division One.
It would need a deduction of 46 or more points to push Durham into a relegation place, thus either reprieving eighth-placed Hampshire from relegation, or giving Division Two runners-up Kent a pathway to the upper tier.
Hampshire chairman Rod Bransgrove told BBC Radio Solent: "I think that if they're penalised, that relegation is the right move for them.
"I don't want to see the back of Durham. I've got no problem with the ECB doing what they need to do to support the situation.
"I do feel, though, that any of us that played Durham when they were being supported were at a disadvantage."
Durham installed floodlights at their Riverside ground in Chester-le-Street in 2015, to bring the venue up to standard for the 2019 World Cup where it will be one of 10 host venues.
But this summer the team did not have an overseas player and as recently as August, chief executive David Harker reassured players and supporters the club had a sustainable future.
An ECB statement read: "It's well known that we've had several conversations throughout the season with Durham CCC about the opportunities and challenges facing the club.
"We are fully committed to supporting the game in the North East and are working closely with the club to ensure a successful and sustainable future."
Durham's England all-rounder Ben Stokes told BBC Sport he was not at the club enough this season to fully understand what was happening behind the scenes.
"There's stuff in the papers, but I don't really know what to take from that," he said.
"I'm hoping that the situation doesn't arise of us being relegated, but all we can really do as players in keep trying to win games, which is what we managed to do, and stay up in Division One."