The 28-year-old is set to complete a loan move to the Bianconeri until the end of the season after being suspended by the Premier League outfit following a training-ground incident
The club's record signing arrived at St Mary's in a deal worth £15 million during the summer, but was suspended for two weeks earlier in January after an incident at the club's training ground.
The Bianconeri confirmed their interest in the Italy international on Wednesday and now look poised to sign Osvaldo until the end of the season, but admitted the club needed to sell before bringing in new players.
One player seemingly on the way out in Turin is Mirko Vucinic, who has been linked with Arsenal and was part of a failed swap deal involving Inter's Fredy Guarin.
"We like Osvaldo and other clubs like him too," director Giuseppe Marotta told La Gazzetta dello Sport. "He is an excellent player, but before buying anyone we need to sell.
"Vucinic? He will leave only if he asks to go."
Southampton manager Mauricio Pochettino hinted at the move on Thursday, adding that "anything can happen" ahead of Friday evening's transfer deadline.
Goal understands the Serie A leaders, after initial reluctance, are prepared to pay a fee to secure the loan deal, which would include the option to buy in the summer.