Hazard, 21, has been attracting attention from most of Europe's elite, including Real Madrid, Manchester City, Chelsea and Arsenal, and has revealed that he is keen to move on from the French champions in the summer.
And Cole, who is currently on loan at Lille from Liverpool, says that he has the necessary qualities to reach the very top.
"Hazard is different class. He's still only 21 and has definitely got another few gears to go up. He's got the same attitude as Rooney at that age - he's fearless," Cole told talkSPORT.
"He knows how good he is, and that's a positive. All the top players have got that belief and a bit of presence and arrogance, and he's like that.
"He's a bit like Messi. There's only one Messi and I don't want to put pressure on the kid, but he's like him - short, squat, powerful legs, great touch, he can finish, he's got everything."
The Belgian midfielder's price has been set at €40 million, but Cole believes he would be better off staying at Lille for now.
He added: "The best thing is for him to stay at Lille for as long as he can. He's playing for a good club, winning titles and there's plenty of time [for him] to go elsewhere. The fans love him, the players love him and we want him to stay."
Earlier this week, Hazard's agent, Marc Formeaux, had said: "People are talking about an offer of €30 million from City and that he would then be loaned back to Lille for free until June.
"His future is with one of European football's prestigious clubs like Real Madrid, Barcelona, Chelsea, Manchester City, AC Milan, Inter or Juventus. For Hazard, the president of Lille wants to receive €40 million while the player himself has declared that he wants to play for Madrid."