The report states that the Istanbul Giants want to make Lennon their next Premier League signing.
Lennon spent last season out on loan at Everton but failed to secure a spot in Roberto Martinez’s starting line-up.
The England international provided three assists and scored two goals in 23 Premier League appearances for the Toffies.
The Istanbul Giants also secured the signatures of Simon Kjaer from Lille, Sener Ozbayrakli from Bursaspor and Josef de Souza from São Paulo. Fenerbahce also signed Abdoulaye Ba and Fabiano from Porto on loan.
Fenerbahce take on the Ukrainian side Shakhtar Donetsk in the UEFA Champions League third qualifying round.