The Turkish side had a £6.4million bid for the Juventus midfielder accepted.
Poulsen was offered a contract worth £3.7million-a-year.
He said: “I am honoured by Fenerbahce’s offer, £3.7m is a lot, but to me it’s not just a question of money. I want to continue with Juventus.
“For me, it’s very important that my family finds itself well in the city in which I play.”