Vietto struggled in his only season at the Vicente Calderon, scoring just three goals in 28 games.
However, at just 22 years old, the young forward is still highly-rated and will look to get his career back on track after impressing for Villarreal two seasons ago.
Vietto's move to Sevilla coincides with Kevin Gameiro's move to Atletico Madrid, thus making the opposite move to Vietto.
The news will come as a big blow to Barcelona who were tracking both players as potential back-up to Messi, Suarez and Neymar.