Icardi ignored a wide-open Osvaldo and instead shot wide on a counter-attack in the second half. Osvaldo then rushed to scream at Icardi, while midfielder Fredy Guarin separated the two.
The 28-year-old did not train on Thursday with the rest of the team, fuelling rumours of an imminent exit.
AC Milan, Torino and even Fiorentina, where Osvaldo played from 2007 to 2009, are immediate possibilities.
Osvaldo is currently on loan from Southampton and it is unlikely that he will return to the Premier League club.
The Italian international scored five goals in 12 Serie A appearances for the Nerazzurri.