Gazzetta dello Sport said the fiery Italy international, who became Southampton's club-record signing when he joined for £15 million from AS Roma in August, would be loaned out subject to a medical in Turin on Friday.
Juve, who have to pay €400,000 (£328,000) for the loan, would also have an option to buy the 28-year-old for €18 million (£14m), Gazzetta added. Sky Sports also said a deal was imminent.
Osvaldo was suspended by Southampton last Thursday for two weeks following an incident at the club's training ground which British media said was a bust-up with team mate Jose Fonte.
Southampton did not disclose what had happened but said in a statement: "The club has taken swift and proper action for what it considers a breach of the conduct expected of its players."
Earlier this month Osvaldo, who has only scored three goals in 12 appearances for Southampton, was fined £40,000 and banned for three matches by the English FA for his part in a touchline fracas against Newcastle United in December.
Osvaldo, who has 12 international caps, has a history of disciplinary problems. He was also kicked out of Italy's Confederations Cup squad last year after throwing a tantrum during Roma's defeat by Lazio in the Coppa Italia final.
TV cameras caught Osvaldo apparently insulting Roma coach Aurelio Andreazzoli after being left out of the starting lineup.
He began his career in Argentina with Huracan and has also played for Atalanta, Lecce, Fiorentina and Bologna in Italy and Espanyol in Spain.