Bob (Robert) Tytler, 63, was last seen in the Hardgate area of Aberdeen on Tuesday, 30 March, however, Police Scotland has said he may have travelled to Edinburgh and the Grampians via public transport.
Police Scotland has confirmed that they have “a number of officers” dedicated to tracing Bob.
Often sleeping rough, Bob is described as white, around 5 ft 7 ins tall, slim build with long grey hair.
He was last seen wearing a black waist length cagoule jacket, green t-shirt, black jeans and black trainers.
Police Scotland is asking members of the public to check their sheds and garages in case Bob sought shelter from the ‘concerning’ cold weather.
Inspector Graeme Moir said: "Bob has been missing for a week now and we are becoming increasingly concerned for his welfare.
"He is known to travel throughout the Grampian area, sometimes further afield, often sleeping rough.
"We believe he may have travelled to the Edinburgh area in the last week using public transport.
"The particularly cold weather is a concern and Bob may not be equipped to deal with the snow.
"I am asking members of the public to check any outbuildings such as sheds or garages they have in case he may have sought shelter from the weather.
Anyone who may have seen Bob or has any information on his whereabouts is asked to contact police on 101, quoting reference number 0812 of 31 March.