90min reported on Friday that City had agreed terms on a deal worth around £20m for the 21-year-old, who will remain in Argentina for the rest of the season before joining up with his new side in the summer.
Alvarez - who has been touted as the successor to Sergio Aguero - had a release clause of around £18m, but it is believed that the two clubs have come to an understanding without needing it to be triggered.
With the transfer nearing completion, The Times have revealed that City's arch rivals United had passed up the opportunity to sign young Alvarez earlier this month.
With the forward's contract due to expire at the end of 2022, the player is said to have been offered to Man Utd earlier in the transfer window, but the Red Devils declined to pursue a deal.