The Citizens have persistently been linked with a move to sign the 20-year-old midfielder, who is understood to have an €8m (£7.1m) release clause in his contract with the Argentine club.
A recent report claimed that Pep Guardiola's side were on the verge of finalising a deal to sign Perrone, fending off interest from the likes of Newcastle United, Wolverhampton Wanderers and Benfica.
However, Velez president Rapisarda has confirmed that they have rejected an offer from City after holding talks with the reigning Premier League champions.
Rapisarda has also referred back to previous negotiations with City involving the sale of Benjamin Garre in 2016, in which Velez complained that the English club were attempting to persuade the midfielder - who now plies his trade on loan at Huracan from Racing Club - to sign before the age of 16.
Speaking to Velez670, via The Sun, Rapisarda said: "I have spoken with the people of City, because we already have a precedent. They handle themselves that way. They were constantly persuading the boy [Garre].
"There is the clause which the players can execute. He should present a note to Agremiados, AFA and the club that is going to execute the clause.
"Yes, the player can authorise a third party to deposit the money. They made an offer, we were not seduced. He is a calibre player."
There is still hope that City can secure the services of Perrone, as Rapisarda has admitted that the youngster wants to move to the Etihad Stadium, while he has also criticised Argentina Under-20s head coach Javier Mascherano for 'persuading him' to join up with Guardiola's side.
"The boy, he wants to go," Rapisarda added. "The City coach called him three times, [Javier] Mascherano also persuaded him, which I don't like.
"The boy loves the coach of that club. We are going to take the risk. We are trying to be the best in the transfer and with numbers that favour Velez."
Perrone made his professional debut with Velez in March last year and has since established himself as a first-team regular, scoring three goals in 33 appearances across the Primera Division and Copa Libertadores.
The midfielder is set to link up Mascherano and the Argentina Under-20s squad ahead of the South American Under-20 Championships, which begins on January 19 and runs until February 12.