West Ham have offered the manager's job to Paulo Fonseca.
The Portuguese is understood to have impressed David Sullivan following a meeting at the Hammers co-owner's Essex mansion.
Fonseca agreed to give Sullivan 72 hours before reaching a decision, with his current club Shakhtar Donetsk keen to extend his contract for another season.
Moyes left the east London side on Wednesday when his contract expired following the final game of the Premier League season.
Fonseca and super-agent Jorge Mendes were involved in talks that lasted for 75 minutes before viewing a luxury one bedroom property in the surrounding area.