The 30-year-old former Forest loanee was released by the Magpies after six years at St James' Park.
He has not played a first team game since the end of his second loan spell at Forest in May 2019.
He first joined the Championship club fox six months in January 2018 before returning to the City Ground for all of the following season.
The former Sunderland player, who has played more than 300 games in the Premier League and Championship, has scored four goals in 55 appearances in all competitions for Forest in his temporary spells.
The club has not disclosed the length of contract Colback has signed.