Moyes has been out of work since he was sacked at Old Trafford in April, just 10 months into a six-year contract, having replaced legendary boss Sir Alex Ferguson.
The Scot recently professed he was ready to get back into work, having snubbed offers since his departure from United.
And that return could now come at Anoeta, with the San Sebastian club's supremo Jokin Aperribay confirming talks with Moyes had been held, as well as former West Brom boss Pepe Mel, to replace the recently sacked Jagoba Arrasate.
He told EFE: "It's true about Mel, it's true about Moyes, but we're also talking with more managers and maybe there will be more than that.
"We are talking with all that are on the shortlist and we want them to know how we operate, giving them knowledge of Real Sociedad, from the respect, to the philosophy and the youth academy.
"We really like the four, each with their own style, and there are some things that we really like a lot about them."
Moyes went toe-to-toe with the Spanish club in his short United reign - beating them 1-0 at Old Trafford in last season's UEFA Champions League group stage before a 0-0 draw in the Basque Country.
