The 42-year-old has been without a job since leaving Manchester United in the summer after the arrival of Jose Mourinho.
Giggs, who served as assistant manger at United for three years, is said to be keen on starting his managerial career and was recently linked with replacing Francesco Guidolin at Swansea.
However, it is claimed the Welshman gave an unconvincing interview and chairman Huw Jenkins instead opted to hire Bob Bradley - who becomes the first American to manage in the Premier League.
But with Giggs continuing to search for his next role, Sunderland legend Gray reckons he could emulate former United team-mate Roy Keane at the Stadium of Light.
“I go back to the years when Roy Keane came in and he was the man who took Sunderland to the Premier League," Gray told talkSPORT.
“He may have had a little bit of a falling out with the players and fans towards the end, but I don’t think you can forget what Roy brought to Sunderland.
“He was the one who got the club back into the Premier League and that was his first job – I think Ryan Giggs would have the same effect.
“I’m not dissing David Moyes here, I’m not trying to get him out of the job.
“All I’m saying is that if there was an opportunity to bring Ryan in, I would take him.”
David Moyes is under increasing pressure at Sunderland with the club failing to register a single league win so far this season and sit bottom of the table.