Edward Gallagher, 78, was last seen at around 10.30am on Monday at his home in Barlia Drive in the Castlemilk area of Glasgow.
The last confirmed sighting of him was at 1am on Tuesday as he got off the number 75 bus and was seen walking towards the Carmunnock bypass.
A man fitting the description of Mr Gallagher was seen on Ardencraig Road at around 1.30am on Tuesday.
Police have issued a CCTV image of the missing man and have appealed for anyone with information to contact them.
Inspector Gary Ritchie said: “Mr Gallagher is very vulnerable given his age and he suffers from dementia. As the time passes his family are very concerned for his wellbeing, particularly during the recent cold snap this week with sub-zero temperatures.
“With this in mind, I’d ask local residents to check their outhouses and sheds as he may have taken shelter in any of these. I would urgently like to speak to anyone who has seen Mr Gallagher to get in touch with police immediately, with any information you have.
“Today we’ve issued a CCTV image of Mr Gallagher on the number 75 bus in the early hours of Tuesday November 22 around 1am, before he left the bus and walked towards Carmunnock bypass.
“We’re concentrating our searches in and around the area where he was last seen, and we’d like to appeal to local residents in the area to help us by keeping an eye out.
“A gentleman fitting the description of Mr Gallagher was seen half-an-hour later in the vicinity of Ardencraig Road.”
Mr Gallagher is white, 5ft, of slight build and bald with a little grey hair at the sides. When last seen he was wearing dark trousers, a dark jacket, a dark skipped hat and tan boots.