Carter, who played 50 games for the club between 1987 and 1992, was found by a passer-by in bushes in the Highfield Close area of Stretford, Manchester.
Emergency services were called but the academy goalkeeping coach from County Durham was pronounced dead at the scene.
Coach Carter dead aged 40
The death of Tim, who also coached the Estonian national goalkeepers on a part-time basis, is not being treated as suspicious police say.
RIP Tim