Mr Walsh, chief executive of BA owner IAG, predicted that the Virgin brand would vanish if US airline Delta succeeds in buying the 49pc stake owned by Singapore Airlines.
The comments came after Singapore confirmed it was in talks over the sale of the stake with "interested parties", which are understood to include Delta.
If Delta succeeds, it has been reported that European partner Air France-KLM may then buy part of Sir Richard's 51pc stake, leaving the entrepreneur without majority control of the airline for the first time.
Writing on his blog, Branson said he was willing to settle the claim with a wager.
"I will pay £1 million to their staff if Virgin Atlantic disappears within, say, five years," he said. "If not, BA pays our staff £1 million."
Mr Walsh reportedly said a deal would spell the demise of the Virgin brand, adding that Delta and Air France-KLM would want full control because owning a large but minority stake had not worked for Singapore, which bought its share for £600 million in 2000.
Mr Walsh and Branson have been fierce rivals ever since Virgin became the first serious challenger to the flag carrier's dominance of transatlantic and Far East routes from the UK.
But Branson, 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.
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.