The 56m (183ft) vessel was carrying 22 people including British, American and Canadian nationals.
Fifteen people were rescued, including a one-year-old British girl, and authorities are continuing their search into the night.
Local media reported the yacht, named Bayesian, capsized at about 05:00 local time after encountering a heavy storm overnight that caused waterspouts, or rotating columns of air, to appear over the sea.
Mr Lynch, known by some as “the British Bill Gates”, co-founded software company Autonomy, which was later bought by tech giant Hewlett-Packard for $11bn (£8.6bn).
Witnesses told Italian news agency Ansa that the Bayesian’s anchor was down when the storm struck, causing the mast to break and the ship to lose its balance and sink off the coast of village Porticello, near Sicilian capital Palermo.
A waterspout is similar to a tornado and can form over oceans, seas or large lakes.
Divers have identified a wreckage 50m below the water’s surface and are searching for those missing.
The director general of Sicily’s civil protection agency, Salvatore Cocina, told the BBC Mr Lynch, his daughter Hannah Lynch and the yacht’s chef were among the missing.
He said the search, involving caving and rescue diving teams, would continue overnight.
The body of one man was found outside of the wreckage.
His nationality has not been confirmed.
BBC Verify looked at corporate records and found the Bayesian’s ownership is tied to Mr Lynch’s wife, Angela Bacares.
Sources close to the matter have confirmed to the BBC Ms Bacares has been rescued.
big disaster’ says captain of rescue boat Fifteen people managed to get to safety after the storm hit.
Ansa news agency reported a 35-year-old mother held her one-year-old daughter in her arms in the sea.
The woman, named as locally asCharlotte Golunski, said: “For two seconds I lost the little girl in the sea, then I immediately hugged her again amidst the fury of the waves.
“I held her tightly, close to me, while the sea was stormy.
Many were screaming.
“Luckily the lifeboat inflated and 11 of us managed to get on board.”
The baby is fine and the mother was treated with stitches, the agency said.
She added she had been on the boat with her husband, who is also safe, and colleagues from a London company.
A doctor based at the Di Cristina Hospital in Palermo, where some of the survivors were taken, said they were “very tired” and “constantly asking about the missing people”.
Dr Domenico Cipolla told Reuters news agency: “We have given the survivors this information, but they are talking and crying all the time because they have realised that there is little hope of finding their friends alive.”
Survivors said the trip has been organised by Mr Lynch for his work colleagues.
In the initial aftermath, a nearby Dutch-flagged vessel rescued survivors from the waves, tending to them until emergency services arrived.
Captain Karsten Borner said after the storm had passed, the crew noticed that the yacht that had been behind them had disappeared.
“We saw a red flare, so my first mate and I went to the position, and we found this life raft drifting,” he told Reuters.
That life raft was carrying 15 survivors, three of whom were “heavily injured”, he said.
Divers search off Sicily coast during yacht rescue operation.
A local fisherman told Reuters news agency he had seen people being rescued by an inflatable boat dispatched from another yacht.
The captain of a local fishing trawler said he saw debris, including cushions from the deck, floating in the sea.
Footage from the wreckage site showed helicopters circling over several coastguard vessels as divers wearing bright orange descended into the water.
Eight of those rescued are receiving treatment in hospital, the Italian coastguard said.
The western half of the Mediterranean has experienced severe storms since the middle of last week.
Through Sunday night and into Monday morning, a clutch of bad weather passed by the north coast of Sicily.
BBC Weather forecaster Matt Taylor said: “A waterspout is a tornado that has occurred over water rather than land.
“They can form during intense storms, on the base of cumulonimbus/thunder clouds.
“Turbulence, and the wind blowing in slightly different directions around the cloud, can cause rotation under the base of the cloud and the spout to form.
“Like tornadoes, they bring powerful winds, but instead of picking up dust and debris they cause a water mist around the column of rotating air.”
The UK Foreign Office said it is supporting a number of British nationals and their families following an incident in Sicily.
Britain’s Marine Accident Investigation Branch is also sending a team of inspectors to conduct a “preliminary assessment” into the sinking of the UK registered-boat.
The Bayesian’sregistered owner is listed as Revtom Ltd.
The superyacht can accommodate up to 12 guests in six suites.
The yacht’s name is understood to be based on the Bayesian theory, which Mr Lynch’s PhD thesis was based on.
Mr Lynch’s wife Ms Bacares is named as the sole legal owner of Revtom registered in the Isle of Man.
A spokesperson for Camper and Nicholsons International, the firm that manages the 2008-built boat, told BBC Verify: “Our priority is assisting with the ongoing search and providing all necessary support to the rescued passengers and crew.”
Mr Lynch sold his company Autonomy to American computing giant Hewlett-Packard (HP) in 2011 for $11bn (£8.6bn).
But an intense legal battle following the high-profile acquisition loomed over Mr Lynch for over a decade.
He was acquitted in the US in June on multiple fraud charges, for which he had been facing two decades in jail.
The sinking of the yacht came on the same day that Mr Lynch’s co-defendant in the fraud case, Stephen Chamberlain, was confirmed by his lawyer as having died after being hit by a car in Cambridgeshire on Saturday.
