Netanyahu said he discussed the operation with Pompeo and called the tunnels a violation of a UN truce resolution that ended a 2006 war between Israel and the Shiite militant group Hezbollah.
Later in a televised address, the Israeli leader also denounced what he called Iran's hand in the construction of the tunnels.
"We are acting with determination to prevent Iran from being rooted in Syria... we are also acting against Iran's terrorist actions in Lebanon," he said.
Netanyahu said he would discuss the matter with world leaders in the coming days and that he was seeking an urgent meeting of the UN Security Council to discuss Hezbollah's "aggression".
US President Donald Trump's White House gave its full backing to the operation, with National Security Adviser John Bolton saying "we call on Iran and all of its agents to stop their regional aggression and provocation".
Hezbollah did not comment in detail but published footage of Israeli soldiers across the border and described the activity as aimed at "discovering tunnels allegedly from southern Lebanon to settlements in northern occupied Palestine".
Israeli army spokesman Lieutenant Colonel Jonathan Conricus said the "attack tunnels" were not yet operational. He declined to say how many had been detected or how they would be destroyed.
The military said it had located one such tunnel dug from a home in the Kafr Kila area of south Lebanon that crossed into Israeli territory and was working to "neutralise" it.
The tunnel runs some 200 metres (660 feet), at a depth of 25 metres, Conricus said, and stretches some 40 metres into Israel.
The area around the Israeli town of Metula has been declared a closed military zone, with the army distributing images of heavy machinery digging into the ground.
Peacekeepers boost patrols
Israel labelled the operation "Northern Shield" and said all its activities would take place inside Israeli territory. No tunnels from Lebanon include exit points within Israel, the army said.
A UN peacekeeping force which monitors the border region said it had increased patrols but noted the area remained calm.
Lebanon's army also said it was monitoring the situation closely.
According to Conricus, the tunnels were part of a Hezbollah plan from 2012 to "shift the battlefield to Israel" and "conquer the Galilee" in a future conflict by infiltrating its territory.
In 2013, the army acted on reports that Hezbollah was digging tunnels, but failed to locate any, he said.
Following the 2014 war between Israel and Gaza, in which the Palestinian Islamist movement Hamas used cross-border attack tunnels, the army said it found "Hezbollah and Hamas share knowledge" and soon after began intensive work to prevent tunnels from Lebanon.
The military has used various means to collapse or fill in tunnels from the Gaza Strip.
Conricus said that while the army has boosted its presence in the north for the Lebanon operation, it has not summoned reserve soldiers.
He said the military "holds the Lebanese government responsible for all activities perpetrated in Lebanon towards Israel".
Netanyahu has spoken of a sensitive security situation without providing details, particularly after defence minister Avigdor Lieberman quit over a controversial Gaza ceasefire last month.
Lieberman's resignation threatened to trigger early elections, but Netanyahu has held his coalition together and is clinging to a one-seat majority in parliament.
The prime minister, who has since assumed the role of defence minister, said at the time that holding elections would be "irresponsible" due to the undefined security threats.
His comments were seen by some as an attempt to save his government, with polls showing wide disapproval of his handling of a Gaza flare-up in November.
Netanyahu is also facing further pressure after Israeli police on Sunday recommended he and his wife Sara be indicted for bribery, the third such decision against the premier in recent months.
Conricus rejected suggestions of politics influencing the announcement.
Israel has carried out hundreds of air strikes in Syria against what it says are Iranian military targets and deliveries of advanced arms to Hezbollah.
However, a friendly fire incident in Syria in September that led to the downing of a Russian plane by Syrian air defences during an Israeli strike has complicated Israeli operations.
Russia subsequently upgraded Syrian air defences with the delivery of the advanced S-300 system.