During Mancini's first spell in charge at San Siro, Inter won three successive Serie A titles as well as the Coppa Italia twice.
Following the sacking of Walter Mazzarri last month, Mancini returned and has won one of his first three matches - a 2-1 success over Dnipro in the UEFA Europa League.
Sunday saw Mancini's men go down 4-2 at Roma, but Zanetti, made vice-president after retiring at the end of last season following 19 years as a player at Inter, believes the club are in safe hands.
"I saw an Inter side that will certainly grow," he told the club's official website.
"We played a great team in Roma, who've been strong for a good while now. We should build on the positives we showed on Sunday.
"I believe that Mancini and the lads will do a great job. As a club, it's our job to be prepared to do whatever it takes to ensure the season is a success.
"There are three matches left until the [winter] break, at which point we'll sit down with Mancini.
"We'll evaluate what's needed in order to ensure that we keep growing."