According to Matteo Moretto, a transfer specialist working with Marca, Alvarez would be willing to take a pay cut to join Real Sociedad and has already asked West Ham United for permission to leave.
The 28-year-old, who can play as a defensive midfielder or centre-back, has a contract in east London until 2028 and was previously linked with Sociedad in July, when reports in the Netherlands suggested Ajax were also monitoring the player.
The chief obstacle to a deal is finance. West Ham paid around €35million for Alvarez three years ago and are keen to recoup as much of that fee as possible, a figure well beyond what Real Sociedad typically spend, while his salary would also need to come down significantly to fit the club's structure.
His injury record is another factor under consideration at Zubieta, with muscular issues having disrupted his recent seasons, a sensitivity heightened by Real Sociedad's own recent experience with Nayef Aguerd.
A deal for Alvarez would address defensive uncertainty that has been on the mind of Yangel Herrera, who spoke to Radio Marca Donostia on Tuesday about his determination to prove Real Sociedad were right to sign him from Girona a year ago, after injuries wrecked his first season at the club.
"I've had a good pre-season, I feel good, calm, physically and mentally, with the desire to find myself again, to have the continuity I couldn't have last year," Herrera said.
"Not to prove myself, but to reaffirm the player I am, and that they weren't wrong to bring me here. That's the excitement I'm feeling."
The Venezuelan midfielder admitted last season had been mentally difficult, arriving as a marquee signing but rarely able to show his true level because of fitness problems.
"It was hard mentally, not being able to show what I wanted to show because of the injuries, but I knew how to handle it and now I'm excited to turn last year around," he said, adding that he needs regular games to hit his best form: "I'm a player who needs continuity to show my best version, I need games under my belt. I'm not someone who comes back from injury with instant sharpness."
Herrera also acknowledged the defensive fragility that has made a player like Alvarez attractive to the club.
"We are aware that conceding a bit fewer of the goals we've been conceding, with the attacking power we have, would help us a lot," he said. "We're trying to find the balance between conceding less and keeping the attacking power we have. Set pieces are one of our targets too, cutting down the goals we give away from set plays."