Duncan McKeith, aged 39, of Cygnet Street, was last seen on George Hudson Street, in York city centre, at around 2.20pm on Friday 21 December 2012.
He has not been seen since and officers are becoming concerned for his welfare.
It is believed that Mr McKeith has family connections in the Sunderland area and it is possible that he has travelled there.
Officers are appealing to anyone who sees Mr McKeith or knows where he is, to contact them with information.
He is described as white, 5ft 10in tall, with a medium build, black hair and hazel eyes.
PC Richard O’Connell, of York police said:
Mr McKeith has not been seen for over 24 hours now and could well have spent the night out in the open in the current adverse weather conditions.
It is important that we establish where Mr McKeith is and his movements over the past couple of days. Therefore, I am urging anyone who has information that can help us, to get in touch as soon as possible.
I would also like to appeal directly to Mr McKeith, to contact the police and let us know if he is ok.
Anyone who can help officers with their enquiries is urged to contact North Yorkshire Police on 101 – select option 1 – and pass information to the Force Control Room.