The 31-year-old is thought to have cost the Turkish giants just €1 million (£800,000), due to a clause in his contract concerning European clubs, and, according to the Turkish Stock Exchange, the Netherlands international will net €2.85m (£2.2m) per season, and €17,500 (£14,000) per game.
Kuyt arrived at Liverpool in 2006 for a fee of around £9m and helped his side to League Cup victory in February 2012, scoring in the final against Cardiff City.
"This is a fantastic club to continue my career," Kuyt stated to the official Sport-Promotion website.
"I really hope to reach a good deal, so I can fully focus again on the Dutch team [and have] no more distraction [with] my near future secured. Subsequently I am all motivated for success with the Dutch team."
"I am ready for a new challenge."
The forward played a less-prominent role last season following the arrivals of Luis Suarez, Andy Carroll and midfielders Stewart Downing and Jordan Henderson, who were all recruited during the 18-month reign of recently-sacked Kenny Dalglish.
And Kuyt was keen to stress that his recent regular demotions to the substitutes' bench had no impact on his decision to move.
“This career move has nothing to do with the Liverpool fans and the club itself. They're forever in my heart," he added.
"It doesn't have anything to do with the new manager [Brendan Rodgers] either, but when Fenerbahce showed me their ambitions, I was interested from the very first minute. They offer me confidence, which I am looking for."
Kuyt racked up 285 appearances during his time in England, netting 71 goals in the process.
The forward has been capped 85 for the Netherlands, and will be representing his country at Euro 2012.