Guler had a €17.5 million ($18.4m) release clause and Los Blancos paid that transfer fee to beat off competition from rivals Barcelona to sign the teenager.
The midfielder has put pen to paper on a six-year deal at the Bernabeu and will be presented to the club on Friday afternoon.
ESPN reported earlier this week that Barcelona had tried to beat Madrid to Arda's signature and had sent sporting director Deco to Istanbul to negotiate a transfer, causing Madrid to up their offer and land the highly rated prospect.
Guler made 35 appearances for Fenerbahce last season, scoring six goals and providing seven assists.
A source close to the negotiations told ESPN that Madrid have been following Guler since the 2021-22 season, with Madrid's chief scout Juni Calafat playing a key part in securing the deal.
Madrid boss Carlo Ancelotti plans for Guler to spend preseason with his new team but ESPN sources close to the player do not rule out a loan move to another team in order to secure regular game time.