Simpson, 21, is understood to have been told he can leave the Gunners this summer after failing to fight his way into the first team. After previous loan spells at Millwall, West Bromwich Albion and Queens Park Rangers, the striker is expected to be made available for transfer.
Both City and Forest missed out on promotion to the Premier League after being knocked out in the Championship play-offs. They believe the England youth international could boost their bids to make it to the top flight
Simpson grabbed 12 goals in 39 appearances at QPR last term and scored twice for the Gunners in the League Cup tie against Wigan Athletic in November 2008.
Blackpool, Celtic, Fulham and Hull City are also believed to be in the chase for his signature.