Giggs 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 has 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 UEFA Champions League.
Giggs is not thought to be a contender to take the position full time, and May - a former team-mate of the Welshman - believes it is too soon for the 41-year-old to take charge permanently and would like to see Borussia Dortmund's Jurgen Klopp or current Netherlands coach Louis van Gaal handed the reins.
"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.
"I think the club will go out and get somebody with experience, with that winning mentality, that attack-minded mentality as well that's been sorely missed this season.
"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 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, City and Chelsea next season.
"He will have the character to overcome all the hype that goes with it (managing Manchester United)."
May believes United are fully justified in dismissing Moyes, as the results under him have simply not been good enough.
He added: "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 change now, or do you make 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."
