Cork has made 125 appearances for Southampton since joining the club from Chelsea in 2011, although he has only started five Premier League games this term.
The 25-year-old, who could make his first appearance in more than a month at Manchester United on Sunday after overcoming an ankle injury, is out of contract at the end of the season but has been offered fresh terms at St Mary's Stadium.
Speaking at a press conference on Friday, manager Ronald Koeman said: "He can stay. The board of the club is talking to him and to his agent and we offer a contract.
"Jack Cork by himself has to say yes or no. We're waiting for that."
Koeman also expressed a desire to retain the services of defenders Ryan Bertrand and Toby Alderweireld, who are currently on season-long loan deals from Chelsea and Atletico Madrid respectively.
Both players have excelled after being given the opportunity to play regular first-team football, although Koeman is aware Southampton could face a battle to keep the pair.
Asked what it would take to ensure Bertrand and Alderweireld stay at Southampton, the Dutchman responded: "First of all, a good contract. And the players need to believe in the future of the club and I think both things we can do. But it's always up to the player.
"If they are looking to bigger clubs, to play Champions League, we know sometimes that's the reason that we will lose players.
"We know that, we know our situation. We would like to grow and we would like to play Champions League, but it's a long way [away].
"[They are] two good players, two young players – Ryan Bertrand now is playing every week, he missed that before. The same question is for Toby - he didn't play every weekend in Spain and now he is showing his qualities.
"We will see what happens, but of course we would like to keep these two players."
Koeman has been impressed with his squad's response to a run of five straight defeats earlier in the campaign, which culminated in a League Cup quarter-final loss to third-tier side Sheffield United.
Southampton have taken 10 points from a possible 12 since that setback and cemented their position in the Premier League's top four with an impressive 2-0 victory over Arsenal on New Year's Day.
"I was very angry [after the Sheffield United defeat] and all the players were disappointed and then it’s an easy job because you need always the reaction of the players and the players showed a good reaction," Koeman explained.
"We changed the system a little bit after the last defeat and we had a good game against Everton [a 3-0 win] which brings a little bit of the confidence back we had before."
