Stewart Leask, 70, was last seen by his wife at their property at Whiting Bay, Arran at about 3.45pm on Sunday.
He went out for a walk towards Glenashdale Falls with his German Shepherd dog Corrie and was expected home a short time later.
Corrie was found safe near Dyemill car park in Lamlash on Monday morning amid a large-scale search to trace Mr Leask.
Coastguard and police helicopters, RAF Mountain Rescue, local mountain rescue teams and volunteers have joined search efforts.
Officers have again urged anyone with information on his whereabouts to come forward.
Mr Leask is described as white, 6ft tall, of medium build with thinning dark-brown and grey hair and a beard.
He wears glasses and when he was last seen he was wearing a red parka jacket, blue jeans and a grey trapper-style hat.
Mr Leask was believed to have been heading from the Whiting Bay heading towards Glenashdale Falls
Insp Christine Boyd said: “Officers have been carrying out extensive searches around the area where the dog was found, the Glenashdale Falls area and also the wider area covering Lamlash and Whiting Bay.
“Due to the terrain and the cold weather conditions, our local officers are being assisted by support unit officers and specially trained search advisors.
“The force helicopter, the Coastguard helicopter and volunteers, RAF Mountain Rescue, as well as local mountain rescue teams are also out searching for Stewart.
“We are extremely concerned for Stewart’s wellbeing and safety. He has had some health issues lately and although he regularly walks in this area he always returns home after a short time. He has never been missing for this length of time.”