Apollon are eager to strengthen their midfield ahead of next season and have watched Djoum in action for the Edinburgh club. They have already contacted his representatives and further meetings are planned next month.
The 30-year-old's contract expires at the end of this month, although he is due to meet the Hearts manager Craig Levein before then for a final discussion on his future.
Hearts want to persuade Djoum to sign a new long-term contract but the player has strong interest from abroad. He has yet to make a decision on where he will play next season.
Reports in Cyprus have confirmed Apollon's interest but other clubs in Greece and Turkey are also competing to sign the Cameroon internationalist.
After playing for Hearts in Saturday's Scottish Cup final defeat by Celtic, Djoum is certain to feature for his country in this summer's African Cup of Nations in Egypt.