The report states that the Black Sea based side have agreed a €3 million per-season deal with the Senegal international.
However, Trabzonspor are yet to agree a transfer fee with Newcastle. The Turkish Super Lig outfit have offered €5 million but the Magpies are holding out for a higher fee.
The 30-year-old joined Newcastle for £9.5 million from Bundesliga outfit Freiburg in 2012.
Cisse has a total of 41 goals and 10 assists in 110 appearances for the Magpies in all competitions since joining.
The Senegalese international struggle for playing time last season but still managed to hit the back of the net 11 times in 22 league games.
Kiss still has another two-years left on his current deal with Newcastle and could be tempted to leave in pursuit of regular first team football.
Beşiktaş are currently manager by former Turkish national team manager Şenol Güneş and finished last season in third place.