"The player told us that he felt it was a one-off opportunity to try his luck in Europe and that he wanted to go, so that is what will happen," said Velez president Fernando Raffaini.
"This is not up to us and we are not doing this because we need the money. We have already offered the player improved terms, but this was his decision."
Fiorentina are expected to pay just over €1.7 million for the player. The 30-year-old Silva was expected to travel to Italy on Tuesday night to complete the deal. He follows former Velez teammates Ricardo Alvarez (Inter Milan) and Maximiliano Moralez (Atalanta) in leaving the Clausura champions for Italy this summer.
Silva is well travelled, having started his career in Uruguay before having spells with Corinthians in Brazil, Energie Cottbus in Germany, and Beira Mar in Portugal. He first moved to Argentina with Newell's Old Boys in 2005, and has also played for Gimnasia La Plata and Banfield as well as Velez.