Moyes was appointed the Sociedad boss in November 2014, but according to The Mirror, he is a managerial candidate for West Ham.
Former Liverpool defender Carragher, though, believes that the Scot should remain at the Spanish club.
“Moyes would be better served having another 12 months with Real Sociedad,” the 37-year-old wrote in The Daily Mail.
“It is a great experience for him taking on Real Madrid and Barcelona and that will also give him more time to put his Old Trafford departure behind him.
“The United job is just like managing England: scrutiny follows those who have held the post and, if Moyes was to come back to West Ham and lose a few games early on, the attention on him would be greater than if another man was given the job. Neither he nor West Ham need that right now.”
Carragher does make a valid point. Moyes’s time at Manchester United ended in disaster, and he will come under scrutiny if he returns to the Premier League next season.
Moyes is a very good manager - his work at Everton proves that. Perhaps the Scot needs to have a successful time at Sociedad before coming back to England.
West Ham have lofty ambitions and want to make themselves regulars in Europe in the coming years. Moyes is probably not ready for that at the moment.