Fenerbahce announced the transfer to the Turkish stock exchange and on the club's Twitter account.
Liverpool then added in a statement: "Liverpool Football Club can confirm that Dirk Kuyt has advised the club that he has today reached an agreement with Fenerbahce SK over a move to Turkey, and the transfer is expected to be completed in the next few days."
Kuyt departs Anfield after a six-year spell that saw him become a firm fans' favourite thanks to his tireless performances on the right flank.
Though initially signed as a striker from Feyenoord for a fee of £9 million in 2006, Kuyt moved into a wide role and leaves Liverpool having scored 51 times in 208 Premier League appearances.
During the 2011-12 season he scored only twice in 22 league starts as Liverpool finished a disappointing eighth under Kenny Dalglish, who this week was replaced by Brendan Rodgers.
However, Kuyt did win the Carling Cup to secure his first piece of silverware with the club.
"After six years I am ready for a new challenge," Kuyt said on the website of his management team.
"(The club and supporters) I will always carry in my heart.
"The appointment of the new coach has nothing to do with it. But when Fenerbahce came to me and made their ambitions known, from the first moment I was sold. They give me the confidence that I lost."
Kuyt has won 85 caps for Netherlands and will represent the country at the Euro 2012 finals which begin next week.