Team GB coach Stuart Pearce is able to select three over-age players in a squad otherwise comprised of under-23s, but Lineker does not believe that the LA Galaxy star should be among them.
"If you are going to pick the best three over-age players then you would say no [to Beckham]," he reportedly told BBC Radio.
"If it is a 'thank you' for all the work that Beckham has done for the Olympics and getting it to London, then it is a different matter.
"I would pick my best three players if I was in charge. That is who I would want, but who knows? It might be a done deal."
Beckham, who has previously expressed interest in captaining Team GB, is among a shortlist of 80 players in contention to play in the summer tournament.
The draw for the tournament will take place on April 24.