The 31-year-old's future at Ibrox is uncertain and Apollon are hopeful of luring him to the Mediterranean.
Lafferty's contract runs until summer 2020 but he has started only seven matches under manager Steven Gerrard this season.
Since leaving Hearts for Glasgow last August, he has scored seven times in 30 appearances for Rangers.
Apollon have declared an interest in taking him to the Cypriot First Division and are waiting to see if he is surplus to Gerrard's requirements for next season.
If so, they hope to sign him and Hearts midfielder Arnaud Djoum to strengthen their squad for a tilt at the league title next season.
They can offer both players the carrot of European football having finished third in the Cypriot league to earn a place in the Europa League qualifying rounds.