The Argentine club's president has said they are keen to secure a deal for the troublesome forward.
The 29-year-old has been on loan at his boyhood club since February, after a similar agreement at Serie A side Inter Milan ended in turmoil when he fell out with club officials.
He has been in decent form for the Argentinian outfit, scoring seven goals in 15 games, although a permanent move seemed unlikely after the club ruled out trying to sign him, according to the Mirror.
However, the club’s president has now hinted at a complete turnaround, saying they are keen to make a deal happen.
“We will talk with Southampton because we would like to keep him,” he said.
“We will ask, or renewal of the loan, or exchange with other players. It will be important also to know what is the will of the player.”
Osvaldo remains Southampton’s record signing after joining from Roma in 2013 but it has been a far from happy stint for him with the Saints, with a training ground bust-up with captain Jose Fonte seeing him leave on loan to Juventus.
That was followed by the incident at Inter Milan, with trouble seeming to follow the Argentine forward wherever he goes. He has scored just three times in 15 appearances for the Saints, who will likely be keen to sell should Boca make a suitable offer.