Osvaldo has been on-loan at Inter from Southampton.
"At the moment I don't know, it's something the player is pursuing," Ausilio told Inter Channel.
"If he's happy to go there for five months, the last word will go to Southampton since he's not owned by Inter, but we're happy to do that.
"We gave him the freedom to find the destination that feels right for him, with our full support. If he goes back to Buenos Aires and it makes him happy, then it makes me happy.
"He won't come back after four months in Argentina, because he's not ours.
"He's loaned to us until June 30, on July 1 he'll become a Saints player again."
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