The Argentina striker was signed by Atletico last year from Manchester City and has proved a massive hit in his 18 months in Spain.
Griezmann recalled at a Hyundai event: "I knew the club was doing everything they could to convince him, so I decided to write him a ten or twenty-sentence message to convince him.
"A message in which I told him all the good things about Atlético, Cholo, and his teammates... And finally, after much persistence, he arrived. I think I wrote to him about four times before he finally made a decision."
On last season's sudden form collapse, the French veteran added: "I had so many hopes, so many dreams, and March wasn't a good month for anyone; we lost La Liga and the Champions League.
"And my morale plummeted, it hurt me a lot mentally, and that affected my game, it hurt me a lot mentally, and that affected my game later."