The 23-year-old has been told by the Spanish club that he can leave this summer and sides from around Europe are interested.
Southampton are keen to take the Argentina attacker on loan, while he is also attracting the attention of Marseille, Sampdoria and Villarreal.
And now, according to Marca, Monaco have also thrown their hat into the ring to land Vietto.
The French champions could well be in the market for a new forward this summer as Kylian Mbappe looks set to leave.
The teenager is wanted by a whole host of clubs – although Paris Saint-Germain appear to be the frontrunners.