Osvaldo was hit with a two-week suspension by the club for an altercation with team-mate Jose Fonte.
Saints' turbulent start to 2014 took a nasty turn last week when Osvaldo was handed a two-week ban by the club for a training ground incident, which Press Association Sport understands saw Fonte head-butted.
The incident looks like it may well have been the last straw for the Italy international signed from Roma in the summer, with a return to Serie A with Inter Milan or Juventus mentioned.
On Monday afternoon Saints boss Pochettino faced the media for the first time since the altercation and proceeded to bat away numerous questions about the incident, repeatedly saying there would be no comment on the matter.
The former Argentina international also refused to confirm whether there had been any interest in Osvaldo, before eventually relenting a little on his no-comment stance.
"This is something that is in the past," Pochettino said. "We like to see the future.
"What is done is done and now we are fully focused on concentrating on what is better for the team in the future.
"It is clear it is not a nice situation, but not only here, it happens everywhere in the world. But, as a manager, I have to look for what is better for our club, to look forward."
Asked if Osvaldo's departure before the January window closed would be best for club, Pochettino added: "We don't know."
Pochettino was more forthcoming when quizzed on more general transfer activity in the last few days of the window.
Having previously suggested there would be no acquisitions, there now looks likely to be movement before Friday's deadline.
"We are working on certain aspects and, again, we will inform in the moment that this is done, if it's done," he said.
"What we are assessing is the balance of the squad. Obviously, we have had people working with this and maybe we will see something in the next few days."
As well as bringing players into St Mary's, the headlines will no doubt continue as to whether they can hold onto Luke Shaw.
The 18-year-old left-back is one of the most-sought after English talents, with Manchester United and Chelsea reportedly prepared to go to battle for his signature.
"At this moment, Southampton have no interest in selling Luke Shaw," Pochettino said.
"Not in the summer, either. Southampton have got a lot of expectation put in Luke's future and we have no intention of selling him.
"What is clear is that Southampton believe in these players, want these players, and we have no intention of selling them."
Shaw, who himself has repeatedly reiterated his desire to remain at the club, is all but certain to start in Tuesday's Barclays Premier League clash with league leaders Arsenal.
Victor Wanyama could return from a hairline leg fracture to feature in the match, although Rickie Lambert will be assessed ahead of the league leaders' visit.
"Arsenal have got a great structure," Pochettino added on Tuesday's visitors to St Mary's.
"They have got a squad with very talented players and the structure of the club allowed them to fight for the title until the end of the season. They will be there fighting for the title."