The striker joined Southampton in a £15million deal from Roma last August, but his time in England has been fraught with problems.
Osvaldo was fined £40,000 and banned for three matches last January after an allegation of violent conduct during an away trip to Newcastle United was proven to be true.
It went from bad to much, much worse for the short-fused Italian when later that same month he allegedly headbutted teammate Jose Fonte in a training ground bust-up.
The Saints suspended the forward, who only scored three in thirteen games, for two weeks and eventually shipped him out on loan to Serie A champions Juventus, where he spent the remainder of the season.
The Argentinian-born player was supposed to report back for pre-season training along with the rest of the Southampton squad, but reportedly failed to show and the club have since given him a leave of absence while they attempt to find a suitor.
Southampton's Dutch manager Ronald Koeman told BBC Radio Solent: “It was a difficult situation for him to come back to.
"He was too late. The decision of the board was to allow him not to come back. We have to look for opportunities for him to play in a different team.”
A return to Italy has been widely reported as the most likely destination for Osvaldo, and Napoli and Inter are both said to be interested in acquiring his services, but only on a loan deal.
With his temperament Southampton may struggle to offload him permanently this summer, and even if they succeed they will not get near what they paid for him.