Fenerbahce will pay Benfica 22.5 million euros for the winger, with additional performance-related payments that could take the total value of the transfer to 25 million euros, the Portuguese club added in a statement.
Benfica said Galatasaray, Akturkoglu's former club, had been informed as they hold a preferential right on the player.
Fenerbahce were eliminated from the Champions League after a 1-0 defeat by Benfica in their playoff second leg on Wednesday, with Akturkoglu's goal decisive for the Portuguese side.
The 26-year-old winger joined Benfica last year after winning consecutive Super Lig titles with Galatasaray.
Earlier on Friday, Fenerbahce announced that they had parted ways with manager Jose Mourinho.