The Istanbul Giants have agreed a €9 million fee for the Danish International with Lille. If Fenerbahce are win the Super Lig title in the 2015-16 season they will pay Lille an extra €1 million.
Kjaer has been offered a four-year deal with Fenerbahce at €4 million per season. The 26-year-old will also receive appearance bonuses.
Southampton have also been linked with the former Wolfsburg centre-back as a replacement for Toby Alderweireld.
The Istanbul giants are planning to strengthen their squad ahead of next seasons UEFA Champions League play-off round.
Fenerbahçe finished the league season as runners up and booked a place in the Champions League play-off stages.
Kjaer scored three goals in 42 appearances for Lille in all competitions this season.