The England international, 22, has been linked with a move away from the Riverside Stadium since the end of last season with Everton the latest to show an interest.
But Southgate told the club's official website: "Stewart is such a good player and is now an established player in the England squad, so I'm not at all surprised other clubs might be interested in him, but that's as far as it goes.
"Stewart is definitely not for sale so other clubs would be wasting their time even considering making an offer. The idea we would want to sell the likes of Stewart is way off target."
Vi bruger cookies for at kunne give dig den bedste oplevelse. Ved at bruge vores side accepterer du brugen af cookies. Du kan læse mere i vores persondatapolitik.