Southgate said: "I felt it would have been good to have retained Mark's experience because he was always a good example for younger players.
"With that in mind, we made him a fair offer, but he decided to take up a different offer and move on.
"I enjoyed playing alongside and managing Mark because he was always an outstanding professional.
"He was a very consistent goalkeeper for us for more than 11 years, so he deserves great credit for that, though the club has looked after him too.
"I have wanted to see what Brad and Ross are capable of and they will now get the opportunity to show me.
"They will come back for pre-season training on an equal footing with the first-choice goalkeeper's shirt up for grabs. It's up to them to take that opportunity."