The ex-Red Devils captain came up with the punishment after Carragher wrongly predicted that former Tottenham Hotspur midfielder Paulinho would score 15 Premier League goals in a season.
Carragher initially refused to wear the top of his arch rivals, but after Neville suggested that he should auction the shirt off for Alder Hey Children's Hospital, the retired defender accepted.
Alongside a picture of one of his old number two shirts, Neville tweeted: "Morning @Carra23 . Here is a potential deal- Post a pic in the shirt/ sign it and auction on here for Alder Hey!Deal?"
Carragher eventually responded by tweeting: "Anything to do with Alder Hey has to be done!"