Gardner joined the Black Cats from Birmingham for £6m last June, but has struggled to settle in the area, leading to speculation over his future.
West Brom manager Roy Hodgson declared his interest after hearing of Gardner's troubles, but O'Neill is keen to keep the midfielder at the Stadium of Light.
"Gardner can score the goals," O'Neill told Sky Sports News. "I don't think that anybody wants him to leave the football club.
"I know that homesickness can have an effect on you, but he has come up trumps again.
"He scored today and scored a magnificent free kick against Wigan a couple of weeks ago. The crowd have enjoyed him."
Sunderland are currently enjoying the joint best form in the Premier League, with only Manchester City able to match their 13 points from the last six games.