The striker is contracted with Villarreal for another five years but had been strongly linked with a move to the rival La Liga club this summer.
Rossi's agent Federico Pastorello said that Barca had failed to meet Villarreal's €30 million (£26.5 million) demand for the player and discussions have now ceased.
"Barcelona never offered €30 million, they offered less. Villarreal wanted more and so in the end there was no agreement," Pastorello told Radio Mana Mana. "In the end, [Barca] have gone for Sanchez, who will join them for around €40 million.
"At the moment the situation is now closed. We will see what happens. Rossi wanted to join Barcelona, and so I won't exclude he will stay at Villarreal another year."
He added that the possibility of a move to Juventus was "remote", confirming that there had been "no contact" from the Italian side.