The former Manchester United midfielder left Old Trafford last year and claims Premier League clubs Arsenal and Chelsea, along with AC Milan, were interested in his signature.
But the France international was only interested in joining Juventus and he wants to go on and win lots more silverware with the club after lifting the Serie A crown last term.
The 20-year-old told Sky Italia: "Since I was a child I have always liked Juventus, I was a fan of them. For French people this club mean a lot.
"Sir Alex Ferguson tried to keep me at Man Utd: also Chelsea, Arsenal and Milan wanted me, but I had only Juve in my mind.
"I'm happy to be here and I want to win a lot of trophies with this jersey."
Pogba scored his first international goal in France's 4-2 World Cup qualifying win over Belarus on Tuesday and he would love to double up with a goal at the San Siro, although he acknowledged it is the result which counts.
"I'm happy to have scored my first goal with the national team, even if it seemed easy to me," he said.
"I'd like to repeat myself at the San Siro too, although against Inter winning is all that counts."