Patrick Harvey, from the Kilsyth area of Lanarkshire, was last seen in Glasgow at about 10pm on Saturday, July 2.
Police believe the 69-year-old boarded a train from Glasgow Central Station to Blackpool at around 9.30am the following morning.
Patrick is described as 6ft 1in tall, with greying hair.
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He was wearing blue jeans, black jacket and grey trainers at the time of his disappearance.
Sgt Pamela McFarlane, of Cumbernauld Police Station, said: “Our enquiries are ongoing to trace Patrick.
“It is out of character for him to be away for so long and his family are growing increasingly concerned for his welfare.
“We have had reported sightings of him in Blackpool but we now believe that he is in the Bolton area.”
Anyone who has any information about Patrick’s whereabouts should contact police immediately by calling 101, quoting incident 1051 of July 4.