The 22-year-old had a disappointing maiden season at the Vicente Calderon and is not in Diego Simeone's current first-team plans.
Sevilla and Barcelona are both battling it out to sign the Argentinian and it appears that the Andalusian club has won the race.
Vietto is said to be keen on working with Sevilla's new manager Jorge Sampaoli and securing first-team football at the Camp Nou would be a struggle, given Barca's front line of Messi, Suarez and Neymar.
Barcelona have now turned their attention to AC Milan striker Carlos Bacca.