Thomas Bates, 30, was last seen in Tesco Superstore, Chepstow, at around 9.30pm on March 2.
North Worcestershire Police said officers are concerned for his welfare.
He is described as 5'9" tall, of slim build, with a shaved head and tattoos of song lyrics and phrases on both arms.
Thomas was last seen wearing a grey coat with fur around the hood, a light grey hooded jumper, black jogging bottoms, white trainers and a baseball cap.
Anyone with any information on his whereabouts is asked to call police on 101.