He gave Carragher a signed shirt with a special message for Neville written on the back.
Carragher has got Liverpool new-boy Xherdan Shaqiri to sign a Liverpool shirt for Neville to wear, after the pundit was critical of the Swiss global in the past and it remains to be seen whether the duo will now swallow their pride and wear the shirts on air.
Earlier this month, Carragher defeated Neville in a penalty shoot-out, which meant the fullback would have to wear the jersey of his most hated rival.
"That's a Liverpool shirt - I'll even buy it", Carragher said.
Neville heavily criticised the Swiss midfielder during the 2018 World Cup before he completed his transfer from Stoke City to Liverpool.
Liverpool visit Crystal Palace on Monday night, with Carragher and Neville due to take part in their first MNF show of the season.
Neville responded to Carragher's tweet - but, seemingly, he didn't appreciate the gift. "Please don't track back", Shaqiri wrote, alluding to Neville's belief that he doesn't do enough to help out his teammates when defending.
Whether or not Neville now intends to honour his bet and make amends by wearing Shaqiri's Liverpool shirt in front of a live television audience of millions remains to be seen.