Having scored 20 goals for Ajax in the 2016/17 season, Klassen signed for Everton in the summer for a £24m fee. The Dutchman's arrival heralded the Toffees' ambitious transfer plans, but it ended up being a disappointing season for both.
Klassen made just three Premier League starts and a total of 13 appearances across all competitions for Sam Allardyce's side. The former England boss criticised the midfielder on a number of occasions and he looked certain to depart Goodison Park in the summer - although that could change now Allardyce has been sacked.
Nevertheless, as per the report, Besiktas have approached Everton with an offer for the 25-year-old. The Black Eagles, who finished fourth in the Super Lig, want to take the Netherlands international on an initial loan deal, which could later become a permanent.
The decision as to whether he remains in Merseyside will likely be left to Everton's new boss Marco Silva. Meanwhile, Klassen recently revealed his desire to stay with the Toffees and fight for his place in the team.
"My overall experience at Everton has been good. The club is excellent and the people are very nice, it is an ambitious club and I 100 per cent enjoy being here," he insisted, as quoted by the Liverpool Echo.
"I feel at home and am determined to play more for the team. We have a vacation now and I will relax for a couple of weeks but then start training to come back in top condition. That is my goal and everybody's goal, so there is going to be good competition for places."