Carroll has joined on-loan from Liverpool.
Brady wrote in The Sun: "Andy Carroll is a gentleman and a fascinating conversation reveals that he is desperate to stay in the England squad.
"It is a privilege that carries no appearance money with it, because Andy reveals the England players pool their fees and hand them to their charity, Club 66.
"Obviously, that is a reference to the World Cup triumph of 1966.
"I have never heard that before and I think it is a wonderful gesture, something the players should be applauded for at a time when their riches often earn them criticism."