World record signing Bale's future has been the subject of speculation after becoming a target for angry Real fans this season.
The Wales forward's car was damaged following the Clasico defeat to Barcelona last weekend and United legend Scholes hopes the 25-year-old will opt to return to the Premier League by joining Louis van Gaal's side.
The former England midfielder wrote in his column in The Independent: "My feeling is that he would be perfect for Manchester United,
"Of course, as a fan of my former club I would like to see him there but, looking at it objectively, I always felt that Bale was a natural fit for United as he developed at Tottenham from a left-back into one of the best attacking footballers in the country.
"He would be loved at Old Trafford. In return I believe he could turn Louis van Gaal's team into serious title contenders next season. In my years at United I witnessed some signings who, over their careers, transformed the fortunes of the team.
"From Eric Cantona when I was an apprentice, to Dwight Yorke, Ruud van Nistelrooy and Wayne Rooney. These were great footballers who became great United players. Bale has the potential to make as much of an impact as any of them.
"All good players need to be appreciated. Bale would be coming to United in a time of change when the team is looking for leaders. Madrid will always be in a state of upheaval, with new presidents and a fresh intake of players each summer.
"At United there has also been great change in the past 22 months, but the club is trying to move towards stability in the post-Sir Alex Ferguson era."
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