The Saints' record signing had fallen out of favour at St Mary's Stadium following a training-ground incident in which he reportedly headbutted team-mate Jose Fonte, which saw him suspended for two weeks, and he will now continue his career at Juve as he aims to play his way into Italy's World Cup squad.
"I'm very happy," Osvaldo told his new club's official website. "It's all been a bit of a whirlwind and I still can't quite believe it but I'm happy.
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"It's been a pretty hectic three or four days but I'm happy."
Should Antonio Conte's side make the transfer permanent in the summer, the deal will be worth €19 million (£15.6m) over three years.
The 28-year-old joined Southampton from Roma at the start of the 2013-14 campaign and scored three goals in 13 Premier League appearances.
The striker previously enjoyed spells in Italy at Atalanta, Lecce, Fiorentina, Bologna and Roma, while also wearing the jerseys of Huracan and Espanyol.
He has a contract with Southampton until June 2017.