The England striker, who netted twice against Norwich on Saturday, says the 40-year-old has “all the credentials” to succeed sacked David Moyes on a long-term basis
Following David Moyes' sacking on Tuesday, the 40-year-old has been placed in temporary charge at Old Trafford until the end of the season, and he guided United to a commanding 4-0 victory over Norwich City on Saturday.
"It was a bit strange [playing under Giggs]," Rooney told Sky Sports.
"I've played with him for the last 10 years but we all know Giggs is a really experienced player and has been part of the coaching staff for the last season.
"He's built to be a manager. Us as a team, we can see that. We can see by the way he's handled himself since he's took over. It was fantastic to get out there and get the win for him.
"I think Giggsy's got all the credentials to be the next boss but that's down to the board, that's down to Giggsy. Who knows?"
Rooney opened the scoring against Norwich from the penalty spot after 41 minutes and doubled United's lead shortly after the break. Juan Mata, who started on the bench, also struck twice and the Spaniard is happy to be working under Giggs.
"For me it's a pleasure to play under his management," said Mata.
"When I was a kid, I remember him playing. He's a proper Man United boy. He was born almost in here and I think for all the players it's a pleasure to be with him in the training ground and every game.
"But it's not down to us, the decision. We have to now enjoy with him until the end of the season."