In a tweet, Israeli military spokesperson Avichay Adraee said the Lebanese government is responsible for the build-up of the tunnels, saying they are endangering Lebanese citizens.
An Israeli military digger works on the border with Lebanon in Metula on Tuesday.
Within a few hours of the original announcement, the IDF said it had located a tunnel in Kfar Kela near Metulla, which sits right on the Lebanese border. He declined to say how many were detected, when they reached Israeli territory and how they would be destroyed.
He said that the operation would not cross the border into Lebanon.
Operation Northern Shield "is not expected to spark an escalation" between the IDF and Hezbollah, according to Haaretz, which adds, "The question is how Iran will respond".
Since then, Hezbollah leader Hassan Nasrallah has touted plans to take over parts of Galilee in northern Israel in a future conflict.
He said a task force established in 2014 has been searching for potential Hezbollah tunnels for more than two years.
'MISSION COMPLETE': Photo of George H.W. Bush’s Service Dog Goes Viral
Mr Bush , who served as the 41st USA president between 1989 and 1993, died late on Friday at the age of 94. Martin's Episcopal Church in Houston , where his late wife, Barbara, was memorialized earlier this year.
The latest operation comes a day after Israeli Prime Minister Benjamin Netanyahu met US Secretary of State Mike Pompeo in Brussels, where he was reported to have been updated on the anti-tunnel action.
"Whoever tries to harm Israel's security will pay dearly", Netanyahu said in a statement.
There was no immediate comment from Hezbollah and its Al-Manar TV did not make any mention of the Israeli operation. The official, who spoke on condition of anonymity in line with regulations, had no further details. "The army command is ready to confront any emergency", it said.
Meanwhile, Lebanese president Michel Aoun on Monday morning received US Ambassador Elizabeth Richard to discuss the current developments in southern Lebanon.
There has been increased attention in Israel in recent days over Israeli officials' concerns regarding Iranian activity in Lebanon.
Before his departure, Netanyahu hinted at the Israel operation.
Israel also faces terror tunnels in the south from the Hamas terror organization in Gaza.
Israeli officials have long warned the threat posed by Gaza's Hamas rulers pales in comparison to that of Lebanon's Iran-backed Hezbollah - a heavily-armed mini-army with valuable combat experience and an arsenal of some 150,000 rockets that can reach almost every part of Israel.