Like with Ross Turnbull, skysports.com understands a number of clubs are showing an interest in him, including West Ham, Bolton and Everton.
"Ross has turned down two or three contract offers and has said he wants to wait until the summer, which is his prerogative, and Matthew's similar," said Southgate.
"Matthew's been a difficult one because up until two or three months ago he hadn't played a game after coming back from three cruciates.
"Again, Matthew's been made an offer, he's also said he wants to focus on his football for now, which I can understand, and we would like them both to stay.
"I think Matthew in particular in recent weeks has shown his value to us. Obviously the club, certainly the top people at the club, looked after Matthew very well during his injuries.
"We hope that he'll bear that in mind when he's thinking about what he wants to do. It is still very early for Matthew in terms of his recovery from the injury, but he's proving himself to be a good player for us.
"We realise the situation that we're in that young players are going to have more opportunities than they may elsewhere. These are two are slightly different scenarios but Matthew's certainly one where you hope there's a lot of good will with what's gone on and Ross, I'm surprised he hasn't signed anything yet, but that's his decision."