The Saudi-registered Phenom 300 plane was trying to land at the airport when it crashed on to dozens of vehicles at a car auction site at just after 3pm on Friday, sparking a huge blaze.
The four victims have not yet been named, but the Saudi ambassador to Britain offered his condolences to the Bin Laden family.
The statement said: “His Royal Highness Prince Mohammed bin Nawaf Al Saud … has paid his condolences to the family and relatives of Mohammed bin Laden at Blackbushe Airport in Britain for the great loss they have suffered as a result of the crash of the plane that was carrying the family.”
Another statement said the embassy would work with the UK authorities to ensure a quick repatriation of the bodies for burial.
Footage uploaded to Twitter showed fire and smoke billowing from the crash site, with the flames coming from the middle of a car park where empty vehicles had been lined up.
Acting Chief Inspector Olga Venner of Hampshire Police said: “We can confirm that there were four people on board, including the pilot.
“Sadly there were no survivors. No-one on the ground has been injured and we would urge anyone with any information, including pictures or videos, to contact 101.”
A group of people, believed to be the family of the victims, were escorted by police to the site, where the scorched remains of the jet could be seen lying beside burnt out cars.
The jet had reportedly flown in from Milan, Italy.
The Bin Ladens are a large and wealthy family who own a construction company in Saudi Arabia, although the surname is most commonly associated with Osama, the terror leader who masterminded the attacks on the US on 11 September, 2001, and was shot dead by US special forces in 2011.