The England Under-21s midfielder made his Gunners first-team debut in 2007 aged just 17 but failed to make more than a handful of appearances for the club.
The 21-year-old also enjoyed loan spells at Scunthorpe, Watford, Norwich City and West Ham, helping the latter to promotion last season and new Forest chairman Omar Al Hasawi was full of praise for Lansbury, who becomes Sean O'Driscoll's seventh summer signing at the City Ground.
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Lansbury could be in line to make his debut against Charlton on Saturday.