The Black Eagles were linked with Cisse last season before they signed Demba Ba from Chelsea.
The Newcastle based newspaper Chronicle has stated that the Magpies have set a €14 million price tag on the Senegalese international.
Cisse provided one assist and scored 11 goals in 22 appearances for Newcastle this season. The 30-year-old joined Newcastle from SC Freiburg for €14.5 million back in January 2012.
The Premier League striker has also previously played for Metz, Chateauroux and AS Douanes.
Beşiktaş finished the league in third place and Bilic resigned from his position at the end of the season before joining West Ham.