Vieira has been in charge of Manchester City's elite development squad this season and had been touted as a possible candidate for the first-team job if Manuel Pellegrini departed.
However, there reportedly are some concerns that lacks the necessary experience to become manager at the Etihad.
And that has seen Marseille step up their interest in the World Cup winner, according to The Telegraph.
It is claimed that OM will look to capitalise on Vieira's ambition to bring him to the Stade Velodrome as a replacement for Marcelo Bielsa.
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Vieira is close to being formally qualified for a first-team job after spending the past two years securing his coaching licence with the Football Association of Wales.
And despite remaining a highly-regarded member of the City staff by the club's owners, the 46-year-old may now be on his way back to France.