Southgate wants O'Neil, Boateng at Middlesbrough
On Boateng, Southgate said: "He's part of our squad here and a very strong part.
"I can't guarantee any player a first-team place, we're trying to create a club where people are pushing for places.
"He may have perceived that he'd dropped down the pecking order and found that difficult but I think he realises what I'm trying to do with the squad and the number of games he played speaks volumes."
O'Neil was an automatic selection following his move, but off-the-field issues have raised major concerns about his long-term future at The Riverside.
However, Southgate insists the player is feeling better about life on Teesside.
"He's said to me he feels more settled and at this moment in time I have to go along with that and hope the situation resolves itself over the summer," he told the club's official website.
"Gary is contracted to us and he's someone we don't want to lose."