The 28-year-old arrived in a club-record deal from Roma in the summer, although has failed to live up to the billing.
Allied to that, the Italy international's relationship with many team-mates has become irreparable after a training ground altercation with Jose Fonte.
Osvaldo was handed a two-week club suspension for the incident and looks all-but certain to leave before the transfer deadline, with Juventus the front-runners for his signature.
Pochettino said there will be no problem if a move fails to materialise, but admitted the forward has burned too many bridges to feature again.
Asked if it would be possible for Osvaldo to play for Saints again after the incident, Pochettino said: "Your question answers itself, if you know what I mean, meaning no. It's clear."
On the situation, he continued: "We have to wait [to see whether he goes]
"There is still a day and half left and he is suspended at the moment and we shouldn't get ahead of ourselves.
"As soon as anything happens about this matter the club will make it public.
"It's clear this is something that no one expects. It's completely unpredictable and something we did not expect. In light of that there is no explanation I can give."
Pochettino said he was ''not in charge of that situation'' but was able to confirm that Saints had turned down a bid from Swansea for Jack Cork.
The former Argentina international also suggested the need to bring in a new face has lessened somewhat thanks to Sam Gallagher's impressive senior debut in midweek.
Despite failing to score, the 18-year-old led the line with aplomb against Arsenal - a performance that has seemingly earned Pochettino's trust.
''It's clear that on Tuesday Sam Gallagher played and played very well against Arsenal, a product of our academy and our team so why not give him the opportunity to keep playing?'' he said.
''I am not saying that [we will not sign anyone] but signing right now someone who fits the profile we are looking for is not easy. It's very complicated to find that player.
''With Sam Gallagher's performance against Arsenal on Tuesday it made us confident we might give him a chance and keep counting on him because he showed he can compete at the top flight.''