His current contract at Everton runs until the end of the season and it seems unlikely he will sign a new one with the Blues, with Hasenhuttl keen to keep him at St Mary's.
He told Hampshire Live: "I think in the games he played he has shown that he was an important part of our game - especially his workload and effort.
"He's a very professional guy and, as I always say, I was happy he was coming here and I still think he's a player that can help us in the next few years.
"If he's not always playing from the beginning, then he's still such a good character that he supports the team whenever he needs to. I think this is for me a very important part for giving him a new contract or not."