Bale has been in fine form for Spurs recently, scoring eight goals in his last six games, leading to speculation linking him with the world's top teams this summer.
A win for Tottenham in the North London derby tomorrow will put them in a strong position to qualify for the Champions League next season, but Keown believes that that may not be enough for Bale.
"Tottenham want to get into the Champions League. Of course, they want to have bragging rights but first and foremost they want to get into the Champions League and that is up for grabs," Keown told Press Association Sport.
"If Tottenham are going to keep Gareth Bale they will have to fulfil all of his aspirations for what he wants to achieve and get into the top four. Someday they might have to deal with him wanting to leave if they don't start winning things.
"If Arsenal win they will be one point behind Tottenham and that is what Wenger will be focusing on. Certainly, being in the Champions League helps with money and is a draw for signings in the future. Arsenal will want to keep progressing as a football club. That should not be taken for granted."
Tottenham currently sit in fourth place in the Premier League, four points ahead of Arsenal.