The 21-year-old defender had hoped to put pen to paper during a visit to England last week, but having agreed personal terms, the transfer was put on hold when his medical revealed concerns over a ligament injury suffered last season.
Peruzzi was subsequently sent to see specialist Professor Pier Paolo Mariani at the Clinica Deportiva Villa Stuart in Rome, and was given a programme to address the problem.
He is due to return to England when the Black Cats, who are currently in Hong Kong for the Barclays Asia Trophy, and owner and chairman Ellis Short head back to the north-east.
A statement on Velez's official website, www.velezsarsfield.com.ar, said: "As reported earlier, during the week directors Baldomar and Ordonez travelled to England with the player Gino Peruzzi to advance negotiations over the latter's transfer to Sunderland.
"After agreeing contractual terms, the player underwent a medical examination.
"The professionals found repercussions from the torn ligament Peruzzi suffered last April against Olimpo in the Copa Argentina, which prompted a three-week absence from games.
"To obtain further information about the injury and its evolution, they made contact with Dr Ricard Coppolechia, who answered their questions.
"After this, Sunderland requested permission to subject Gino Peruzzi to a final evaluation at the Clinica Deportiva Villa Stuart in Rome, which is directed by the renowned Professor Pier Paolo Mariani.
"The evaluation had a positive result with Prof Mariani recommending specific work on the affected muscle area.
"Meanwhile, Sunderland had travelled to Hong Kong for some international matches, so the English club's authorities communicated their intention to sign the contract on the return of their president in the course of this week.
"Our club's entourage returned to Argentina on Sunday, anticipating that Gino Peruzzi will travel to England again in the coming days."
Peruzzi would become the latest addition to a much-changed squad with manager Paolo Di Canio having already made nine summer signings and allowed Simon Mignolet, Titus Bramble, Matt Kilgallon and Ahmed Elmohamady to leave the club.