The 30-year-old has been back-up to Lee Camp, but it is understood he wants to leave the City Ground in a bid to earn first-team football elsewhere.
The former Charlton trainee signed for Forest for £500,000 from Southampton in 2006 and made nearly 150 appearances.
Smith played once in the league and three times in the League Cup for Forest last season.
He had initially signed a two-year contract with the club in December of last year.
Smith made history in 2007 when he became he scored the quickest goal by a goalkeeper in world football - netting after only 23 seconds.
In a League Cup tie at The City Ground, local rivals Leicester City allowed Forest to take the kick-off and for Smith to run up and score the goal.
The show of sportsmanship came about as Forest were leading the initial tie 1-0, which was abandoned due to the collapse of Leicester's Clive Clarke.
Leicester went on to win the game 3-2.
Source: BBC Sport
Source: BBC Sport