The Italian head coach was understandably heartbroken after his side fell 3-0 to a clinical Barcelona team at the Camp Nou, leaving their European dreams in tatters.
However, Conte, as always, was gracious in defeat and still approached Messi to praise him on another incredible performance in which he said: "One [player like him] is born every 50 years".
Following the exchange, the manager had no problems revealing what was said to reporters after the match: "I think when you have the opportunity to give a compliment to a person like Messi, it's fine to do it, it's the right thing to recognise when someone is a super super player".
He added: "We are talking about someone who scores 60 goals every season. I am happy to have been able to tell Messi how well he played. I repeat, he is an extraordinary players, the best in the world".
Messi found the back of the net twice against the 'Blues' - his first in the opening few minutes and his second to kill off any hope of a Chelsea comeback - as the Argentine reached 100 Champions League goals, quicker than rival Cristiano Ronaldo.