Alvarez joined the Black Cats on loan from Inter Milan last season, with the deal stipulating Sunderland would pay an £8million fee to sign the 27-year-old on a permanent deal if they remained in England's top flight.
However, after a disappointing season, which saw Alvarez make only five starting appearances in the Premier League, it is understood Sunderland are eager to back out of the deal. Inter are also reluctant to have Alvarez on their books.
The Chronicle is reporting Alvarez could now be headed to San Lorenzo, provided FIFA grants his request to be made a free agent.
"We agreed that when his legal status is resolved we will sit down to define contract terms," San Lorenzo president Matias Lammens said.
"He is a player that is to our liking and we are very optimistic."