The 27-year-old is out of contract with Zenit on June 30, making him an appealing option for Marseille.
OM are having to balance the books, which could lead to €40m-rated defender Boubacar Kamara being sold.
But with Champions League qualification guaranteed, Andre Villas-Boas is keen to add players to his squad.
And Kuzyaev, a 21-cap Russian international, is viewed as a possible replacement for Morgan Sanson, who could be heading to England.
Villas-Boas coached Zenit from 2014 to 2016, leaving a good impression but also giving him contacts at the club.
Kuzyaev can play anywhere across the midfield and his versatility would add depth to the Marseille squad.
Zenit signed Kuzyaev from Terek Grozny in 2017 for €4m, but haven’t been able to renew his contract.
Competition from Spain and Germany
According to L’Equipe, Kuzyaev is eager to experience one of Europe’s ‘Big 5’ leagues, preferably with a Champions League club.
However, Marseille’s place in next season’s Champions League could give them an advantage.
With very little money to spend, Villas-Boas will request Marseille president Jacques-Henri Eyraud act quickly to sign Kuzyaev.