Giggs, who was installed onto Manchester United's coaching staff by Moyes in the summer, could be forced into hanging up his boots if he is not offered a new one-year deal before the end of the season.
The 40-year-old signed a new one-year deal at the beginning of March last year but is yet to begin negotiations on a new contract at Old Trafford.
Giggs has become a bit-part player at Manchester United this season as the Welshman has fallen behind the likes of Marouane Fellaini, Michael Carrick and Darren Fletcher in the midfield pecking order.
The Welshman is said to have become slightly disillusioned with life under Moyes as he has appeared in just four games for Manchester United since the turn of the year.
Giggs is expected to stay on Moyes' coaching staff if he is to end his playing career at Old Trafford as the former Everton boss is keen on utilising the Manchester United's experience within the dressing-room.