America's largest airline, Delta, is believed to have approached Singapore Airlines about buying its 49% stake, with Singapore confirming today that it is talking to "interested parties".
If Delta succeeds, it has been reported that European partner Air France-KLM may then buy part of Sir Richard's 51% stake, leaving the entrepreneur without majority control of the airline.
Sir Richard, who set up the airline in 1984, is said to have been weighing its future for several years, appointing Deutsche Bank two years ago to examine offers.
Singapore said today: "Singapore Airlines wishes to announce that it is in discussions with interested parties concerning the possible divestment of its 49% shareholding in Virgin Atlantic.
"These discussions may or may not result in a transaction. Singapore Airlines will make further public announcements as necessary."
Delta and Air France-KLM came close to a deal in February 2011 but discussions broke down over price and Sir Richard's rights over the Virgin brand.
Virgin, which has its headquarters at Gatwick Airport, has a fleet of 40 aircraft and flies around six million passengers a year to long-haul destinations. It posted a pre-tax loss of £80 million in the 12 months to the end of February.
Singapore paid £600 million for its stake in 1999.
Earlier this year, British Airways expanded as it bought BMI British Midland, the second-largest Heathrow operator, giving it more than 50% of the crowded airport's take-off and landing slots.
Delta is America's biggest carrier by the number of passengers, carrying about 160 million a year compared with United Airlines' 140 million.
A statement from Virgin said today: "Virgin Atlantic confirms Singapore Airlines has made an announcement relating to the possible sale of their stake holding in the airline.
"This is a matter for Singapore Airlines and Virgin Atlantic will not comment further at this time."