The attacker - who spent the first half of the season on loan at Inter, but is no longer needed at the Giuseppe Meazza side after a number of incidents - had already been linked with a move to his native country for several weeks and he arrived in Buenos Aires on Saturday to finalise a deal.
Boca have now announced that terms have been agreed and that the deal will be made official on Tuesday.
"Osvaldo will sign his contract on Tuesday," London told reporters. "Once he has signed, we can officially call him a Boca Juniors player."
Boca are expected to take Osvaldo on loan until June, with the option of extending it by a month should they reach the latter stages of the Copa Libertadores.
The Bombonera side hope to make the deal permanent with a two-year contract following the end of the initial loan spell.
Osvaldo, 29, was a club-record €15 million signing for Southampton, but was shown the exit by the Premier League club after a physical altercation with Jose Fonte during a training session.
He previously enjoyed spells with Huracan, Atalanta, Lecce, Fiorentina, Bologna, Espanyol, Roma and Juventus.