Savage was a Wales teammate of Giggs between 1995 and 2004, while he was part of United’s FA Youth Cup-winning team in 1992 alongside Giggs, but left the club without making a first-team appearance.
Giggs had a season as a player-coach under Moyes, including a spell as caretaker manager after the Scot’s dismissal in 2014, then became assistant manager to Louis van Gaal for two years before departing the club in the summer.
“I think they should have appointed him after David Moyes. That’s when I’d have given him the job,” Savage told BT Sport.
“He had four games at the end of the season [after Moyes left] and I think he should have been given it. But he wants to be his own guy now.
“People say: ‘You tout him around because he’s your mate.’ It’s nonsense, absolute nonsense.”
Giggs holds the record for most United appearances, playing 963 times, having joined the club’s academy on his 14th birthday.