According to Tuttomercatoweb, Nice offered €10m and €12m to the Greek side, with both bids rejected.
Olympiacos want at least €20m, a fee Nice don’t want to pay anything close to.
The two clubs could meet in the middle at €15m, but that is a feeling rather than a certainty.
The Greeks are in no hurry to sell 24-year-old Tsimikas to Patrick Vieira’s side, knowing he is a valuable asset.
Olympiacos hope to start an auction in the €20m range, but Nice want to close the deal before it can reach that stage.
Tsimikas has made 27 appearances for Olympiacos this season and is a three-time Greek international.