The 22-year-old Redcar-born defender has been linked with several top-flight suitors, including Stoke, following Boro's failure to win promotion from the Championship at the first attempt.
However, with Emanuel Pogatetz and Chris Riggott having left the club and manager Gordon Strachan still working to make loan signing Stephen McManus' stay permanent, Gibson insists he does not want Wheater to go.
He said: "David Wheater is exactly what this club wants.
"He is totally committed to this club, he is a great lad and if you ask him to do something, he will do it. He is Middlesbrough through and through.
"I think there's more to come from David. Fabio Capello included him in his England squad not that long ago. Since then, he's experienced a lot of change that hasn't helped him.
"Ultimately, I have an influence, but I don't pick the squad, Gordon Strachan does. But Gordon loves him.
"David's conduct around the place doesn't suggest he wants to go and he's not the kind of lad or player that we would want to leave this club.
"I want David Wheater at Middlesbrough Football Club."
Source: Team Talk
Source: Team Talk