The 27-year-old, who joined the Saints in a £12.8m deal last month, said that he is happy to have linked up with Mauricio Pochettino's squad.
"Was I in exile? No, I didn't feel that way," Osvaldo told reporters. "I decided to leave along with the club, who made it clear they wanted me to go.
"The normal fans always showed affection for me. There is an element of the fans who love football the wrong way and I could not stay in Rome with people writing nasty messages outside my house.
"I wanted to try an experience abroad, Southampton wanted me and so I decided they were the best option for me. I am really happy. I need to adapt to their style of football, which will take a little time, but I have faith."
Osvaldo is currently on international duty with the Italy squad.