The ex Manchester United and Everton manager, who has also been linked with the vacant US national team job, is interested in taking over when the current MLS campaign ends in December.
Atlanta, who are currently second in the Eastern conference, play in the new $1.6bn Mercedes-Benz stadium and have the league's highest average attendances – regularly over 50,000 fans, under billionaire owner Arthur Blank.
Martino's Atlanta contract expires at the end of the campaign, and although he has an option to extend it he has given no hints over his future.
But it is understood Moyes is the club's first-choice should the 55-year-old former Argentina manager decide to move on and talks have already taken place.
Moyes, 55, has been out of work since guiding West Ham to Premier League safety last season.
He parted ways with the London club, who later appointed ex Manchester City boss Manuel Pellegrini, after just six months in May.
Atlanta's technical director and vice-president Carlos Bocanegra is a huger admirer of the Scot, and has received glowing endorsements of his methods from his former US national team-mates Tim Howard and Landon Donovan, who both had successful periods under Moyes at Everton.
Former centre-back Bocanegra also played for Rangers in the Scottish Premier League, where Moyes, who played for Celtic as a defender, has long been linked with returning to manage.
A source close to the former Man United manager told The Independent that Moyes had always loved America, and that the first-class facilities and forward-thinking nature over there appeals to him.
The project at Atlanta, with their state-of-the-art new stadium and talented squad, is understood to appeal strongly to the Scot.
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