After returning to Italy as part of a deal that included Saphir Taider going the other way, the former Roma and Fiorentina forward explained his eagerness to play regular football again after failing to settle at St Mary's Stadium and an unsuccessful loan spell at Juventus during the second half of last season.
"It's an honour, I'm very happy, I can't wait to start again," Osvaldo told reporters at his official unveiling as an Inter player on Thursday.
"I'm so hungry. The last year has been particularly hard. I really want to go back to my usual standard."
Inter could make Osvaldo a permanent member of the Nerazzurri squad next season for a reported €7 million (£5.6m) fee but the Italian does not see this alleged valuation as a sign of decline following his €19m (£15m) transfer to Southampton last summer.
The Argentina-born striker added: "I do not see this as a negative depreciation. If I do well, I'll have a chance to stay at Inter for many years.
"[Last season] I do not know what happened. I have not adapted to English football. Then I moved mid-season and I have not had continuity - rightly so because of [Carlos] Tevez and [Fernando] Llorente [at Juve].
"I do not look back, I look forward and think to hit the targets with Inter."
The 28-year-old also spoke of his excitement at working under Walter Mazzarri and possibly playing alongside Mauro Icardi.
"Mazzarri and I will talk in the coming days but it is a pleasure to be trained by those who wanted me in the past," Osvaldo enthused.
"[Icardi] is a player with extraordinary qualities, I really like him. Having never played together I cannot say whether we can play together, we'll see."