The former Manchester United trainee played a key role in helping Hull back into the Premier League but is facing an uncertain future after already rejecting two offers this year.
And while Hull have offered Chester, who is soon entering the final 12 months of his existing deal, a bumper new contract, the terms appear to have fallen short of the 24-year-old's expectations.
With Aston Villa and Newcastle both reportedly monitoring the defender's situation over the summer, the Tigers' managing director Nick Thompson remains hopeful over Chester's future at the KC Stadium.
Speaking to the Hull Daily Mail, Thompson said: "We're talking with James' agent and the offer we've put forward is one we feel is good and appropriate.
"It's a tough negotiation because we have some gap between where the two parties are. There's a disparity between where we think the right figure is and the expectations of the player's agent.
"I'd certainly like to think it's something we can resolve. James was virtually an ever-present for us last season and a very important part of our team."
City have been trying to resolve Chester's future since first deferring a contract extension to 2016 last year, and Thompson added: "If you get into the final year and you can't agree on anything then you're going to make other clubs aware of the situation.
"We'd very much like him to be with us next season and we want to come to an agreement where everyone is happy."
And the managing director hopes for fewer complications with out-of-contract trio Liam Rosenior, Paul McShane and Mark Oxley.
"We've made contract offers that we believe are good and fair, and demonstrate our belief in the players," he said.
"I'm sure their agents will be looking at those deals and weighing up their options but we'd like to get those deals wrapped up before the end of June."