The 24-year-old is set to leave the Ligue 1 club after two successful seasons in France, where he recorded 48 goals and 17 assists in 81 appearances.
Marseille had initially valued Greenwood at €50 million but have now agreed a deal worth €40 million (£34 million) plus bonuses with the Turkish side.
Despite interest from clubs including Roma and Atletico Madrid, Greenwood has approved the move to Fenerbahce.
The Super Lig giants are also holding discussions with Manchester United, who are entitled to 40 per cent of the transfer fee.
Greenwood is expected to sign a four-year contract worth around €10 million (£6.5 million) annually.