Keane, speaking at ITV Sport's Euro 2012 launch this week, said: "I think it's only right if Chelsea win that John Terry gets up straight away and lifts it for the first time. He's captain of the club and deserves it."
Keane said United never even asked whether he would be involved in the trophy handover - it was not until 30 minutes later that he and Paul Scholes got their hands on the cup.
The former Republic of Ireland midfielder added: "We lifted it half an hour later and it was nice but I don't think United did anything at all [to ask].
"I don't think it crossed our minds at all. The priority was for the team to win and the bonus is always lifting the trophy so I thought don't worry too much about it.
"But I wouldn't begrudge John Terry the chance if they win on Saturday to pick the trophy up."