Moyes could become a second successive former Premier League manager to take over at the Türk Telekom Arena after ex-Manchester City boss Roberto Mancini stepped down last month.
Should Gala appoint the 51-year-old, he will become the fourth Scotsman to manage the club after Billy Hunter, George Dick and Graeme Souness. Twelve Englishmen have managed the Istanbul club, including Malcolm Allison and Don Howe.
Moyes was sacked less than 10 months into a six-year United contract having presided over the club's worst season in 25 years and was replaced by Louis van Gaal in May.
He spent 11 years in charge at Everton previously but has never won a major trophy in his career.
Mancini guided Galatasaray into second place in the Super Lig behind Fenerbahçe and won the Turkish Cup but came to a "mutual agreement" to terminate his contract.