Wenger has been Gunners manager since 1996, but the Frenchman has come under pressure in recent weeks as their Premier League title challenge has stalled, while they have also been knocked out of the Champions League and FA Cup.
Reports in the Italian media subsequently suggested that former Manchester City boss Roberto Mancini has been identified as a potential candidate to take over at Emirates Stadium should Wenger move on.
But Zanetti has now denied the rumours, telling Mediaset Premium: "The club has been clear and Roberto is keen to remain with us.
"Now we must focus on the end of the season, which will be very tough, but I think we have what it takes to move forward."
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Inter were leading Serie A at the mid-season break, but erratic form in the second half of the season has seen their title ambitions fade, and they now sit fifth.
The Milan outfit are also five points adrift of Roma, who occupy the final Champions League space, but Zanetti remains optimistic of securing third place.
"We believe we can finish third," he added. "The draw at Roma gave us a strong response and great confidence.
"The lads have been doing well for the last month and even if we are five points off Roma, I say anything can happen."