The 33-year-old former Spain international, who joined Real's youth setup in 1985 and made his first-team debut a decade later, helped Real win five domestic league titles and three European Champions League titles.
A hero among hard-core fans and a club captain, he would be coached at Besiktas by German Bernd Schuster, who was in charge at Real between 2007 and 2008.
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"I am leaving with the knowledge that the Real Madrid family loves me and that's very important," Guti said at a news conference at Real's Bernabeu stadium on Sunday.
Asked about his future, he added: "I have an interesting offer from Besiktas but I haven't decided yet. What I know for sure is that wherever I go I will give my best."
Meanwhile another of Real's long-term servants, striker Raul, is also set to leave the Bernabeu and will announce his departure for Bundesliga club Schalke 04 on Monday.
Raul, 33, another club captain and a talisman for the fans, made his debut for Real in La Liga in October 1994 at the age of 17 and has won six league titles and three Champions Leagues, twice finishing top scorer in the Spanish league.
In October last year, he equalled German Gerd Mueller's record of 66 goals in the three main European club competitions when he scored against AC Milan in the Champions League.
He holds the all-time scoring record for Spain with 44 goals in 102 appearances, one more than his successor in the number seven shirt David Villa.
