AS reports there are only minor details to agree and the Scot is expected to be at Anoeta to see the visit of the champions.
Talks were finished on Friday and a deal that runs until 2017 is on the table, although it could be structured to run until the end of the current campaign with an optional extension.
Moyes stands to become the best-paid coach in the history of the club with a salary in the region of €2m (£1.5m) a year.
There won’t be an announcement before the game against Atletico to ensure focus is as much as possible on the match.
Moyes is in San Sebastian and should take training on Monday. He won’t be assisted by former England defender Phil Neville, however, as Neville has opted to continue working in the media.
Instead, Moyes’ will be joined by former Liverpool and Scotland midfielder Gary McAllister, it has been suggested.
The make-up of Moyes’ back-room team will be confirmed in due course, AS add.
La Real are in the market for a new boss after sacking Jagoba Arrasate following a poor start to the season, that included elimination from the Europa League before the group stage.
Moyes, meanwhile, left Manchester United in April, having previously spent more than a decade in charge at Everton.
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