Press Association Sport understands the 30-year-old Argentina international has not submitted a formal transfer request, but has indicated his desire to return to South America for personal reasons.
That has alerted the likes of San Lorenzo, who upped the stakes on Friday when president Matias Lammens told ESPN Radio Rivadavia: "Time is running out for the signing.
"When you go for a class A defender, the timings are a bit different, but if in the next week we don't have a response then we will have to go for another option."
Lammens' comments will infuriate the Magpies as they attempt to persuade Coloccini, who signed a four-year contract extension in March last year, that his future remains at St James' Park.
The club has not commented on rumours that Coloccini wants to leave after they resurfaced on Friday. His father Osvaldo revealed San Lorenzo's interest in the player several days ago, and they have made little secret of their intent to sign him despite admitting they cannot afford to buy him.
Lammens added: "Coloccini's arrival is very difficult. We can't go out and buy him; we have always said that we depended on the desire of the player to rescind [his deal].
"We can offer him a very good contract in the case of him rescinding, but San Lorenzo have nothing left to do with regards to the negotiating."
Coloccini cost Newcastle £10.3million when they signed him from Deportivo la Coruna during the summer of 2008. He struggled initially to settle into English football and was part of the squad which was relegated at the end of his first season.
However, he has since rebuilt his reputation to the extent that he was handed the armband by manager Alan Pardew following Kevin Nolan's departure for West Ham and led the club to fifth place in last season's Barclays Premier League.