The Black Sea Storm are reported to be using fellow Swedish international Erkan Zengin to convince him to make the move.
Zengin is reported to have made an enquiry regarding whether he would consider a move to Turkey in the summer.
The 25-year-old still has a year left on his contract but has not renewed his current deal yet.
Ekdal has four goals and two assists in all competitions for the Serie A outfit this season.
The Swedish international joined Cagliari on a £2.6 million move from Juventus in 2011.
Ekdal has made a total of seven goals and four assists in 118 appearances for the Cagliari since joining.
The 25-year-old has two assists in 13 international caps for Sweden.
Tottenham are also reported to be keeping tabs on the Swedish international.
Trabzonspor are currently sixth in the league on goal difference, sharing the same point total as fifth placed Başakşehir and Bursaspor in fourth position.