Spurs director reveals Bale fears
Mills concedes the appeal of guaranteed Champions League football at the Bernabeu is a strong one, but he has not given up hope of keeping Bale to help fulfil Tottenham's own European ambitions.
Real Madrid coach Carlo Ancelotti revealed last week that the nine-time European champions were in talks with Spurs over a possible deal for Bale, for a reported fee of £86m.
"We very much want Bale to stay," Mills told the Evening Standard. "If a player is desperate to leave, it's very difficult to force him to stay.
"We've seen it in other clubs. Even if he has a contract, you can't force somebody to play for you.
"This summer, we are investing in a great squad and we hope that provides us with Champions League football next year and we start to win trophies, FA Cups. That's what Tottenham need. We have made some good acquisitions."
Andre Villas-Boas has already secured the services of £17m Brazilian Paulinho and club record signing Roberto Soldado for £26m - but Mills insists there is more to come.
"There are others in the pipeline," he added. "Strikers are very high on AVB's list of priorities and Daniel Levy is trying hard to make sure we have strength up front. It's a really exciting time to be at the club."