Andrew, 36, from the Dorset market town of Sherborne, was part of an 11-man crew training on the Artemis Racing Catamaran.
"Shocked to hear of the death of Andrew Simpson, Olympic Champ sailor," tweeted Sir Chris Hoy. "Met him a number of times, great guy. My thoughts are with his family."
The news was confirmed on Thursday.
"It is with immense sadness that Artemis Racing confirms the tragic death of crew member Andrew "Bart" Simpson today in San Francisco," read an announcement on the team's website. "Despite attempts to revive Simpson, both afloat and subsequently ashore, his life was lost."
Andrew, who won a gold medal at the Beijing games and claimed silver at London 2012, was husband to Leah and father to son Freddie, nearly three, who joined him on the Olympic open-top bus victory parade through London last July.
"Tragic, Olympic sailor Andrew Simpson's death in San Fran," tweeted Olympic rower James Cracknell. "Reminder no situation totally safe/controlled. No words enough for Leah & Freddie."
"Such sad news about Andrew Simpson," said Labour MP Harriet Harman. "My thoughts are with his family, friends and all @BritishSailing."
Speaking to the press, Artemis chief executive Paul Cayard said it had been a "tragic day" for the crew.
"The entire Artemis Racing team is devastated by what happened," said Paul, "Our heartfelt condolences are with Andrew's wife and family."
Andrew, who was awarded an MBE by Princess Anne in 2009 also met David Cameron when the Prime Minister visited Weymouth boat yard during the 2012 Olympics.