The Premier League club have lost patience with their record signing, who arrived at St Mary's in a big-money move from Roma less than a year ago.
Striker Osvaldo was loaned to Juventus in January and now look set for a permanent switch following his failure to show up for training ahead of the new season.
"It was a difficult situation for him to come back to," Koeman told BBC Radio Solent.
"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."
Osvaldo scored just once in 11 Serie A appearances last term for Juve, who were crowned champions for the third season running.
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