"We see Hizbollah's activities as a flagrant and blatant violation of Israeli sovereignty and it again proves Hizbollah's complete disregard for United Nations resolutions", said Lt Col Jonathan Conricus, an IDF spokesman.
Israeli Prime Minister Benjamin Netanyahu hailed the early successes of the operation, and warned of a "heavy price" to be exacted from those who attempt to harm the State of Israel.
All operations would take place within Israeli territory, Conricus said, though they were likely to boost tensions with Hezbollah, the Shiite militant group with which Israel fought a devastating war in 2006. The tunnels are proof, he added, that bolsters Israel's long-held opinion that Hezbollah's military strategy includes using civilian infrastructure.
Israel has launched what it calls "an operation to expose and neutralise" tunnels between Lebanon and Israel allegedly made by armed group Hezbollah.
IDF officials estimated the operation would last for a few weeks, but a response from Hezbollah could prolong it further. The group would use firepower and ground units and "the surprise component of that plan was supposed to be tunnels that would allow infiltrators into Israel".
On Tuesday, bulldozers and military vehicles were seen operating near the pastoral Israeli border town of Metula.
"During President Aoun's phone calls, the situation was evaluated in light of the available information about the objectives of the Israeli operation and security agencies were asked to closely monitor the situation", the report said, quoting the official National News Agency. Netanyahu's office said the two discussed "how we can together curb Iran's aggression in the region", including in Lebanon, and that the Israeli leader updated Pompeo on the operation.
Israel's Foreign Ministry spokesman Emanuel Nahshon said Hezbollah is under the direction of Iran.
Conricus said that while the army has enhanced its presence in the north, it has not summoned reserve soldiers. The Iran-backed Shia group has not yet commented on Israel's actions.
Since the start of that war, more than 200 strikes were carried out by Israel according to its military. Amid the crisis he took on the defence portfolio from his resigning minister and promised to pursue a hard line against Israel's enemies during what he described as a "complex" security situation. Northern residents have raised fears in recent years of attack tunnels being dug under the border, spurring the IDF to launch a task force to investigate the concerns in 2014.
Hezbollah is also a powerful political party in Lebanon.
Neither side appears interested in another full-fledged confrontation like the 2006 war, but any skirmish could spark an all-out conflict.