Sport says City are in the running for Griezmann this summer after the Atlético Madrid forward announced he will leave the Wanda Metropolitano.
It's claimed that the Blues, along with Paris Saint-Germain and Bayern Munich, are willing to pay the 28-year-old's €120 million release clause.
"It has been five incredible years, thank you very much for everything, I will carry you in my heart," Griezmann said in the video that was published via Atletico's Twitter account.
"I wanted to tell you that I have made the decision to leave, to face new things, new challenges.
"Thank you very much from the bottom of my heart and see you later."