Cole was Lampard’s teammate for eight years at Chelsea and throughout his entire England career.
He then was managed by his former teammate at Derby last season as they missed out on promotion to the Premier League in the Championship Play-Off final.
Lampard has since taken over Chelsea and has enjoyed early success in the dugout at Stamford Bridge.
He said: “I knew him as a player and as a friend, and then being able to work under him and see the way he works, his ideas and his understanding of the game – it was different to the guy I knew.
“I’m not saying I didn’t think he had it in him, of course he did. I was buzzing to see how he works, the enthusiasm he has coming to work, taking the sessions.
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“For me, I told him: ‘I don’t want you to treat me as your friend. If I’m not training well, give me a rollicking and don’t feel you have to play me’.
“We had that conversation before I joined and he said: ‘Yeah I understand that, and I know how you are as a professional, so that won’t happen.
“It’s just about respect really. The other players who he played with and now he’s managing, players like Cesar Azpilicueta and Willian, I think they understand it too.