It is giving away 240 flights, each matched with a Games ticket, from 20 selected international destinations, including former Olympic host cities Sydney, Moscow and Los Angeles.
The airline, one of the top domestic sponsors of the London 2012 Games, is casting the promotion as an attempt to give British athletes home advantage by boosting the number of Brits attending the Games.
Nominators have to be UK residents although BA's terms and conditions are not specific as to whether nominees have to be British, only that they have to be Facebook friends of the nominator. Winners will be selected by a random draw.
The airline is also offering 50 free flights to friends and family of Team GB and ParalympicsGB athletes, to give them an extra lift when they perform at the Games.
When it goes live on Friday (1 June), the promotion will be supported with print advertising, banner ads and social media, and will run until 14 June.
Luisa Fernandez, British Airways manager of the 'Flying Britain Home' campaign, said: "This is the start of our campaign to get the country behind our athletes and give them an extra boost. Every extra clap, cheer and whoop we can offer, could make the difference to how they perform.
Separately, the airline has made another patriotic gesture by marking the Queen's Diamond Jubilee with a message and logo on one of its Boeing 747s. They were designed as part of a 'Blue Peter' contest by 11-year old Katherine Dewar from Chester and the message reads: "Congratulations, Celebrating Queen Elizabeth II's Diamond Jubilee".
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