However, Beckham, the England captain, will wait until his Real Madrid team mate has time to sign the shirt.
Recalling the scene after Sunday's final whistle, Beckham told reporters: "We said a few things in Spanish and swapped shirts ... He had a smile to say 'that's the way it goes'."
Asked what he would do with the shirt, the England skipper said: "I'll frame it and put it in my house. You have great memories of a great player and memories are good.
"You always look back on things and sometimes it's good to have disappointing memories so you can look forward to the better ones ahead of you."
Paying tribute to Zidane's attitude to the game, Beckham said: "He loves playing and loves training, he works as hard in training as he does in matches and that's what is good to see about players like that. They don't just do it in games.
"I'm proud to be in the same team as him."
Both men faced the responsibility of taking a penalty on Sunday. Beckham's was brilliantly saved
Unfazed by having missed a spot-kick which could have given England a match-winning 2-0 lead, Beckham said he would be happy to keep taking them - even if he kept missing.
NO DEFEAT
"Nothing will defeat me. That's the way I look at things," he said. "Even if I miss the next four or five penalties, then I'd step up and take one. I'm not sure people would want me to though.
"Maybe if I'd taken four and missed four, then I'd let someone like Frank Lampard take it instead - maybe I wouldn't have any choice."
Beckham is now focusing on Thursday's Group B game against Switzerland at Coimbra and channelling the players' fury at having let a priceless victory against the European champions slip away.
"The anger is the good part for the players now," he said. "It's not anger where we want to go out and kick players.
"It's anger where we want to win games, and make people believe in us.
"All the players know that you have to stay concentrated for the whole 90 minutes, and more. It's a lesson for us, but a good lesson, as hopefully it won't happen again."