Inter are among the favorites to dethrone Italian champions Juventus following the arrival of head coach Conte for the 2019-20 season.
The Nerazzurri have already signed Diego Godin, Nicolo Barella, Stefano Sensi and Valentino Lazaro, while Conte is determined to prise Romelu Lukaku from United.
Mourinho spent two years at Inter – winning back-to-back Serie A titles and the Champions League – and the club great told Gazzetta dello Sport: "The Nerazzurri have to win Serie A this season. I'm sure they will."
Inter finished fourth last season, 21 points adrift of Juve under former boss Luciano Spalletti.
Mourinho, meanwhile, hinted at a future move to the Bundesliga as the Portuguese remains out of work.
After two and a half years at Old Trafford, Mourinho was sacked by United in December amid a bad run of form and reports of dressing-room unrest.
Mourinho has been linked with numerous jobs since leaving United, with Benfica, Lyon and most recently Newcastle United said to have been interested.
Asked about his future, Mourinho said: "I really miss football. I miss the adrenaline, the field, my job. Football is football.
"Right now, I am studying German. Bundesliga? Now, I'm studying German because I miss the language. I speak English, Spanish, Portuguese, French and Italian. But I do not exclude anything, not even Germany."