Mancienne is entering the final year of his contract at Forest and Olympiacos hope to persuade Forest into selling the versatile performer.
The former Chelsea man came through the youth ranks at Stamford Bridge but only played six first-team games for the Blues.
The former England Under-21 international enjoyed loan spells at Wolverhampton Wanderers and two years at Queens Park Rangers before leaving Chelsea permanently in 2011 for German club Hamburg.
After three years in Germany, Mancienne returned to England to join Nottingham Forest in 2014 and he has gone on to establish himself as a key player at the City Ground.
Nottingham Forest manager Philippe Montanier
Olympiacos have links with Forest, with the Greek club's president Evangelos Marinakis still in talks to buy a majority shareholding in the East Midlands club, and Mancienne has come up in talks between both parties.
Olympiacos are looking to freshen up their squad before the close of the transfer window after crashing out of the Champions League in the qualifying rounds to Israeli side Hapoel Beer Sheva.
It remains to be seen if Mancienne would consider another move abroad after his spell at Hamburg, but Olympiacos are pushing for a deal for the 28-year-old.