The 32-year-old played alongside Rooney and Gerrard at both the 2006 and 2010 tournaments, though a quarter-final place was the best they could manage in two tilts at the trophy.
But Cole, who completed a move to Aston Villa on Tuesday, feels that a failure to reach the same stage again would see Roy Hodgson's side fall short of expectations.
"My heart says we can win it,” Cole told talkSPORT. "I think from the quality of the squad and from what I hear from inside the camp, anything less than the quarter-finals would be disappointing.
"I have watched all three [pre-tournament] friendlies and I’m very happy. Forget the two draws [with Ecuador and Honduras], I think the performances have been good and Adam Lallana, Raheem Sterling and Ross Barkley all look sharp.
"I genuinely think England have got a massive chance here because those three boys in particular look fearless and Steven Gerrard has got the bit between his teeth too. I’m quite positive about it all."
Rooney especially has come under the spotlight as he heads into his third World Cup still searching for a first goal at the finals.
But Cole believes the Manchester United man will answer his critics and go on to break goalscoring records for the Three Lions.
"Wayne Rooney is proven. He will break all goalscoring records with England.
"The class is there and I just think he’s going to be the difference now. If a ball drops in the penalty box against Italy, there is no other man in England you would want it going to.
"He will come out and deliver this time."