"Christoph thanked us for the collaboration and said he could not refuse the offer because of its very good sports and financial framework.
"We wish him all the best in his new job."
Fenerbahce sacked coach Luis Aragones after the Spaniard led the Turkish club to fourth place in the league, Spanish daily El Mundo reported on Monday.
As well as his previous three-year stint at Fenerbahce from 2003, Daum, 55, has also had two spells with new Turkish champions Besiktas, winning a total of three league titles and one cup before signing a four-year contract with Cologne in 2006.
"My decision is not going against Cologne but in favour of new challenges," Daum said in a statement released through his lawyers.
"I have the chance to play internationally right away which for me as a coach is a major challenge and wish," Daum said.
Fenerbahce have qualified for the Europa League, the successor to the UEFA Cup.
PODOLSKI STUNNED
Daum steered Cologne to 12th place this term after helping them win promotion from the second division the previous season.
He was also instrumental in getting striker Lukas Podolski, a favourite with the fans, back to Cologne for next season.
Podolski, who will return to his home club after an unhappy three-year spell at Bayern, said Daum's departure was unexpected.
"This news comes as complete surprise," the striker said in Dubai ahead of Germany's friendly with United Arab Emirates later on Tuesday.
"But I am confident the club will find a worthy successor and I am certain that the club will meet its targets irrespective of the coach change."
