Respected local paper Diario de Noticias Madeira said Algarve-based Jessiman had been on holiday on the island since the start of the year.
The Falkirk College-educated industrial chemical engineer graduate worked for several British chemical companies before moving into the car industry.
His executive posts including a long spell as General Manager at Volvo Car Sverige before he left Sweden and moved to Belgium in 2007 to become MD at Volvo Cars Gent.
The missing holidaymaker, thought to be aged 60, disappeared about 10:30pm on Wednesday night as he was taking pictures of storm waves near to the Atlantic Palms Hotel in the capital Funchal.
A yellow weather warning was in place for the area when he was swept in the sea.
TV footage of the area where he was last seen showed barriers with a ‘do not pass’ sign on them had been erected to stop people reaching the seafront.
Another sign, with the word ‘danger’ on it, warned people access to the area was strictly forbidden in English and Portuguese.
Dozens of police officers began a land search the night of the incident.
About 60 police, civil protection and navy personnel were involved in a land and sea search today.
Funchal Port police chief Paulo Silva Ribeiro said he was not in a position to confirm the identity of the missing Brit, but said he was based on the Algarve as local reports suggested.
He said: “The Air Force helicopter hasn’t been able to go out today because of the poor visibility, but the land and sea search is continuing.
“The search will continue until at least the end of the day and I will reassess the situation at the end of the day, although my intention is to resume the land search again tomorrow and put a small boat out to sea near the shoreline.
“There’s a possibility he may be trapped somewhere in a cave and we’re searching as best we can, although the sea conditions are making it difficult to do a proper search of the caves.
“But I’m not optimistic. If the worst has happened and he is dead, then there is always a possibility we will not find his body.”
Mr Ribeiro added: “We were alerted around 10:30pm on Wednesday. The missing man was with his partner at the time, who has been spoken to by police.
“Another man was near them who was also swept into the sea, but he managed to reach safety.”
Speaking yesterday as the search entered its first full day, Mr Ribeiro told local media: ““The person who approached the shore probably didn’t realise what the ocean conditions were and we believe a wave came and dragged him into the sea.
“At the time it was the worst stage of the storm of the last few days, with high tides and powerful waves.”
Mr Ribeiro added: “The probabilities of this ending happily at this stage are fairly low, considering the person probably sunk to the bottom and will probably come to the surface only in a few days.
“Nevertheless there is always the chance he is stuck somewhere and we can find him and that’s what we’re trying to do.”