Matthew Dillon, 46, was last seen leaving his home address in Armitage Gardens, Liverpool, on Monday 2 January.
He has travelled to York before, and has visited the city's pubs in the past.
Police are very concerned for his welfare, and urging anyone who sees him or knows where he is to get in touch straight away.
He is white, of large build, with short, sandy hair, and wears glasses. When last seen, Matthew was wearing a blue anorak, cargo trousers and carrying a dark cross-shoulder messenger bag.
Anyone with information that could help find him is asked to call North Yorkshire Police on 101, or 999 with an immediate sighting, quoting reference 12230003853.