The deal for both players is year-long loans, with Osvaldo keen to move back to Italy following a successful spell at Juventus last season.
Italy international Osvaldo, 28, signed from Roma on a four-year deal in 2013 but failed to settle at St Mary’s and was disciplined by Southampton on more than one occasion.
Southampton manager Ronald Koeman said: “It’s part of our philosophy to bring in talented young players who like to learn and develop themselves, and that’s true of Saphir.
“First of all, he is a talented football player. He’s 22 and has already played for Algeria at the World Cup.
“I’ve spoken to him about the positions he can play in. He can play as a controlled midfielder but he can also move up around the box and up front, which is a good thing for a coach
“It was a good meeting because he showed me real ambition and he brings competition to the squad.”
Koeman is still looking to bolster his squad after seeing star players Rickie Lambert, Luke Shaw, Adam Lallana, Dejan Lovren and Calum Chambers all leave.
So far the Dutchman, who joined from Feyenoord, replacing Mauricio Pochettino, has brought in Taider, Graziano Pelle, Dusan Tadic and Ryan Bertrand on loan from Chelsea.