The Argentine forward has been heavily linked with a move to Barcelona in recent months.
"Lautaro isn't leaving Inter," Zanetti told Fox Sports.
"It's not just Barcelona who are interested in him, but Lautaro knows that he's at a great club already.
"He looks happy with life in Italy, which isn't easy.
"I think he's going down the right path.
"For now, he's not leaving.
"He's our present and future.
"There's respect on the clubs' part.
"There are players that have release clauses and there's little you can do when a club comes in and pays it.
"But in Lautaro's case, he's just fine at Inter and I can tell when he's happy."