The Welshman has been placed in temporary charge after United dispensed of former boss David Moyes on Tuesday, just 10 months into a six-year contract.
Moyes paid the price for a poor campaign that has seen United relinquish their Premier League crown with a whimper, while the club are unable to qualify for next season's Champions League, and May believes that the Red Devils should do their utmost to tempt Borussia Dortmund coach Jurgen Klopp.
"I think it will be the wrong decision if Giggsy becomes the manager," he told Perform. "So much as he has been an unbelievable player and an ambassador for United, it's totally different managing and playing football.
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"For me, I'd go for Klopp at Dortmund. I think he's been fantastic but he's ruled himself out already. Managers will do that because they don't want to upset their own club.
"If United go knocking on his door and make a few lines towards bringing him on board then he might change his mind.
"I think, at the moment, [Louis] van Gaal is the favourite - he's got a track record of winning things as well at huge clubs so, going down that line, I think United will be pushing Liverpool, [Manchester] City and Chelsea next season.
"He will have the character to overcome all the hype that goes with [managing Manchester United]."
May believes that United were fully justified in dismissing Moyes as the results under the Scot have simply not been good enough.
He remarked: "I think the results say it all, really. Over the season, to lose 11 games, six at home, it's one of those times - do you make the change now or do you make the change at the end of the season? I think the club have done wisely and chosen to do it before the end of the season."