Martinez, 22, is reportedly being targeted by the likes of Barcelona and Real Madrid after impressing at Inter.
The Argentina international has scored 16 goals in 31 games in all competitions this season.
Zanetti said Martinez was happy at San Siro, where he arrived from Racing Club for a reported €22.7million in 2018.
"I'm happy to talk about [Lautaro] because when we signed him we had been following him for some time," he told Sky Sport on Thursday.
"He was one of the most promising players in Argentine football and [Diego] Milito thought he'd have a big future.
"Now he's an asset for us and he's grown a lot. I see him training every week. He's happy with us and I hope he stays for a long time."
Martinez had helped Inter sit third, nine points behind leaders Juventus, in Serie A when the season was suspended due to the coronavirus pandemic.
Zanetti hopes normal life can resume soon in Italy, which has been hit hard by coronavirus, with more than 13,900 deaths.
"Thinking about recovery is very difficult today. Until April 13 it is right to stay home to allow the epidemiological curve to go down, then we will return to normal life gradually," he said.
"We are in a dramatic situation all over the world. The coronavirus is affecting many populations and should not be underestimated."