Alvarez scored six goals in a midweek Cup game for River.
Gallardo joked afterwards that River would be extremely grateful if City would consider letting him stay until the end of 2022.
He said, “It is difficult for Julián to stay because there is a team that made a great investment, Manchester City takes a diamond in the rough. We are countries that export soccer players, many soccer players have left us in this time and we have tried to survive all those games. Replacing Julián is not going to be easy, we are not going to find a Julián to take his place.
“We have to be aware that it is very possible that he will play the round of 16 of the Cup and that he will go to his new destination, it is so. What happens does not depend on me or what we can do.
“If they want to loan us the player until the end of the year, we have no problem and we will be very grateful. We tell them that we are going to prepare a footballer who is already clearly mature to make the jump, but we could give him even more flight. But Guardiola won't listen to my request. We would be very grateful if you can leave us Julián until the end of the year!"