The Los Angeles Galaxy midfielder had been widely expected to win selection as one of the three over-age players in Stuart Pearce's Under-23 squad for the London Games in July and August.
"Naturally I am very disappointed, but there will be no bigger supporter of the team than me," Beckham, 37, said in quotes relayed by his spokesman on Thursday.
British media reports said Manchester United's long-serving Welsh winger Ryan Giggs, Wales striker Craig Bellamy and Manchester City right back Micah Richards, who missed out on England's Euro 2012 campaign, had been chosen.
An official explanation for coach Pearce's decision has yet to be issued by the FA.
The FA will hand a final team list to the British Olympic Association (BOA) early next week.
"As with every sport, player selection is the responsibility of the National Governing Body, the coaches and performance personnel," a BOA statement said.
"With respect to men's football, the BOA has not yet received from The FA the list of players who are being nominated for Team GB, so we cannot confirm who is, and is not, under consideration for the side. We are expecting the list no later than the early part of next week."
Beckham, who has played for Manchester United, Real Madrid and AC Milan in a glittering career, was watched by Pearce during a recent match for the Galaxy and was included in his initial 35-man squad.
Beckham told reporters last month: "I don't want to be picked on a shirt sale or a stadium filler. I want to be picked because of what I can bring to the team."
Beckham was England's most capped outfield player with 115 appearances but has not played for the national team since 2009.
The world famous trend-setter now misses the chance to captain an England side at World Cup, European Championship and Olympic level.
LOCOG chairman Sebastian Coe said he would now discuss a "Games time role" with the globally recognised sports figure who played a role in London winning the Olympics.
Beckham has been used extensively in the buildup to the event and helped to bring back the flame from Greece.
"David has been an extraordinary supporter, probably our number one supporter, of the Games from the very beginning and is keen to continue his enthusiastic support right to the end," said Coe.
"He really gets this. He is from East London and knows how important the Games and sport are to young people.
"He is a great role model and we are lucky to have such an advocate. I will be talking to him about a Games time role."
As a result of the decision to leave Beckham out of the Team GB football squad, odds have been slashed for him to light the Olympic cauldron. British bookmakers Ladbrokes quoted odds of 5/1.
Critics, who believe football has an uneasy place in the Games because of the vast sums of money professionals earn, have said they would be uncomfortable with the Olympics becoming a "David Beckham show".
The Olympic tournament, originally designed for amateurs but now the only Games event to have an age limit, is not so highly regarded in Europe where the World Cup and European Championship are seen as the elite tests.
Olympic football is popular among South American players, however.
Hosts Britain, taking part in their first Olympic finals since 1960, will kick off their bid for gold against Senegal at Manchester United's Old Trafford stadium on July 26.
The Welsh Football Association, along with the Scots and Northern Irish, have been concerned that the inclusion of their players in a rare British squad for the Games could affect their independence but football's world governing body FIFA has tried to allay their fears.