Created in 1973 and 1974, the artwork was given to the Royal College of Art in London in exchange for rent and auctioned during the weekend at Christie's. The painting had been expected to fetch up to $18.3 million, the BBC said.
This month's London auctions are being watched for signs that the thriving art market may be cooling.
Earlier this year, Bacon's "Study from Innocent X" fetched $52.6 million at Sotheby's in New York -- marking a record price paid for a Bacon piece, the BBC said.
Bacon, who had no formal training as an artist, died of a heart attack in Madrid in 1992.