According to an exclusive from GiveMeSport, the Hammers manager will be out of contract at the end of this season, and Celtic are keen on securing his services.
Moyes has done an excellent job at West Ham this season, and the Premier League club are thought to be keen on offering him a new deal.
However, the Scottish manager is yet to agree on terms with the London club, and it will be interesting to see if the Hoops can snap him up before that.
Celtic chiefs believe that Moyes would be a natural fit for the club because he played for them between 1980-83.
The Hoops will be desperate to win the title back from Rangers next season and need a top-class manager to rebuild the squad. Moyes would be a superb acquisition for them.
There is no doubt that the opportunity to manage West Ham is a more attractive proposition for the Scottish manager. It is doubtful that Celtic will convince the 57-year-old to move to Scotland if the Hammers offer him a new contract.
The former Everton and Manchester United manager is close to guiding the Hammers to European football, and he will be desperate to manage the London club next season.