Four tourists and three crew members were also on the yacht.
Bourgnon went diving on his own on Wednesday, police said, while the four holidaymakers and the instructor had gone together.
The group reported some difficulties in reaching the surface although the area is not particularly dangerous.
The crew members, a cook and a hostess, had difficulty raising the alarm as Bourgnon was the only one who knew to operate the communications system.
A rescue helicopter flew over the area late on Wednesday afternoon but saw no sign of Bourgnon.
The search was due to resume on Thursday.
Bourgnon won the prestigious Route du Rhum transatlantic sailing race in 1994 and 1998.