The departing Borussia Dortmund manager is wanted by Fenerbahce to replace Ismail Kartal.
Fenerbahçe are still in the Super Lig title race and the Turkish Cup semi-finals and will be eligible to play in Europe again after serving a UEFA competition suspension.
The 47-year-old German, who recently confirmed that he will be leaving Dortmund at the end of the season, made the trip with his wife and his agent, Marc Kosicke.
Turkish media have reported that Fenerbahce president Aziz Yildirim is due to meet with Klopp on Monday and give him a tour of the club’s training facilities.
It would come as a blow for West Ham as Klopp was understood to be their No 1 target to take over from out-of-contract Sam Allardyce.
Big Sam is unlikely to be handed a new deal with the club when his contract expires at the end of the current Barclays Premier League season.
Klopp recently confirmed that he will be leaving German club Dortmund at the end of the season.