The 28-year-old joined Southampton for a club-record £15 million from AS Roma last summer but failed to live up to expectations at St Mary's.
Osvaldo made just 13 appearances, scoring three times before being loaned out to Juventus in January after a training-ground fight with defender Jose Fonte.
He scored one goal in 11 appearances with the Serie A winners and is now heading back to Italy to play with Inter Milan, after he agreed personal terms with the club.
New manager Ronald Koeman had confirmed that the club were "trying to help Dani find a new team" earlier this week, after the Argentine-born striker failed to report back for pre-season training at the start of the month.
An initial loan move will prevent The Saints from recouping some of the club-record transfer fee they paid 12 months ago, but the player will not be missed too much after a highly disappointing stint in the Premier League.
His one moment of magic came in the game against Manchester City at St Mary's, as Osvaldo turned the defence before clipping a perfectly accurate shot into the top corner.
Koeman will now look to continue the rebuilding of the squad before the start of the season, with another striker likely to be brought in.
The Dutchman has confirmed that the club are working towards replacing those that have left the club this summer, but has asked fans to be patient.
"We are working on it, our job is to be prepared if someone leaves the club."
"The board knows we need new players. There is no need to hurry. We have a good squad but we would like better players to be competitive."