Moyes wants Giggs to combine a coaching role with his playing duties when the Everton manager takes over from Sir Alex Ferguson at Manchester United on July 1.
Next season is expected to be Giggs's last as a professional, and he's long been linked with a move into management.
And the report claims Moyes wants to utilise his experience of the Manchester United dressing room to ensure a smooth transition once Ferguson retires this summer.
Giggs is United's record appearance maker with 940 and counting so far, and Moyes is keen to draw on his unique knowledge of the club to help ease him into his new surroundings.
The midfielder received his 13th Premier League winners' medal on Sunday following the 2-1 victory over Swansea, and plans to come back for at least one more season at Old Trafford.
But he did admit he's thinking about heading into coaching at some point.
"I'm doing my necessary qualifications, which I'm close to finishing," Giggs said last week.
"I wanted to do that and prepare myself as best I can, if I do go into coaching or I do go into management, when I'm finished."
The report also claims Moyes is eager to keep Paul Scholes at the club following the 38-year-old's decision to retire from playing at the end of the season.
Scholes trained with the youth teams at the Carrington training ground during his first retirement, and Moyes wants him to fulfil a similar role next season.