Ronny Deila stressed his desire to keep Fraser Forster at Celtic following the 4-0 victory over KR Reykjavik.
Executive vice-chairman Ed Woodward insists Manchester United are ready and willing to break the British transfer record.
The versatile 25-year-old, whose current deal was due to expire at the end of the season, has committed his future to the City Ground until the summer of 2016.
"Chris is an ambitious player and wants to play at the very highest level," Forest chairman Omar Al Hasawi told the club's official website.
"We share his ambition for Nottingham Forest and it is fantastic news that he wants to play a key role in the future of our great club."
Cohen has made more than 200 appearances for Forest since joining from Yeovil in July 2007.
He has only recently returned to action after almost a year on the sidelines with a serious knee ligament injury.