Jagoba Arrasate was sacked by Sociedad on Sunday following a 1-0 home defeat to Malaga, which left the team second from bottom of La Liga with six points from their first 10 games of the season.
Following on-going speculation surrounding Moyes's candidacy, president Jokin Aperribay revealed on Wednesday afternoon that the Scot is on a four-man shortlist for the position, along with former West Bromwich Albion manager Pepe Mel, but stressed that the club are taking their time in their search.
'The rumours about David Moyes and Pepe Mel are true, but we are talking to other people as well,' he was quoted in Spanish publication AS.
'The coach who comes in must understand the values and philosophy we hold at La Real, and the importance of our youth academy. He must become steeped in the philosophy of the club.
'The important thing is not to be hasty and to make the right choice. There are things about some of them that we like a lot. The important thing it to fix the position we are in. The new coach must work for la Real, and work hard.'
Moyes has been out of work since leaving United in May, following a tumultuous nine-month reign, and is understood to be open to managing abroad.
He has since been linked to various clubs over recent months, including Crystal Palace and Inter Milan, who are understood to be on the verge of replacing current coach Walter Mazzarri.