The defensive utility is a Bosman target for West Ham and Wigan and Southgate told the club's website: "One of my priorities is trying to get Mathew to re-sign for the club.
"In fact, if we can convince him to stay, I would see him as a key signing.
"He has got the right drive, mentality and attitude - the sort of qualities I want players to have. Plus, the football he played for us during the final weeks of the season was enough to make me feel he can have a real impact for us next season.
"He overcame some serious injuries and the time he has spent out of the game seems to have made him hungry to have made up for the lost time.
"I've been impressed with him and I'd like him to stay."