Chelsea left-back Cole will be making his 102nd appearance for his country and has been given the honour to commemorate receiving his 100th England cap in a game that is part of the Football Association's 150th anniversary celebrations.
Hodgson initially said Cole would only lead the team out before passing on captaincy duties to Frank Lampard, but then later clarified that the defender would retain the armband.
"Everyone is behind Ashley in this matter," Hodgson said. "We talked about it in the past, his reluctance to play the captain's role in front of the mass media, and we all understand and accept that, but everyone thought it would be nice if he put the armband on and led the team out.
"'Frank was more than happy to pass on that honour because we all believe he deserves it. He has been a fantastic servant not only to Chelsea and Arsenal but England as well."
Manchester United striker Danny Welbeck is a major doubt due to a knee injury but Hodgson hopes he may be fit in time to face Brazil in Rio on Sunday.
Hodgson also hinted that Liverpool's Daniel Sturridge is likely to replace Welbeck up front and make his full England debut.
Tottenham right-back Kyle Walker has already been forced to withdraw from the squad after suffering a fibula injury.
Republic of Ireland manager Giovanni Trapattoni has announced his team to face England and, as expected, Robbie Keane starts.
The 32-year-old striker plays after his club Los Angeles Galaxy gave him permission to link up with the Irish squad, although he only flew into London on Tuesday morning.
Trapattoni is without defender Marc Wilson (calf), striker Kevin Doyle (Achilles), winger Anthony Pilkington (knee) and midfielder Paul Green (hamstring) who are all ruled out.
Reading's Stephen Kelly will start at left-back, and Jon Walters will line up on the right side of midfield with Spartak Moscow's Aiden McGeady on the left.
Ireland team: Forde, Coleman, O'Shea, St Ledger, Kelly, Walters, McCarthy, Whelan, McGeady, Keane (capt), Long.