The military has used various means to collapse or fill in tunnels from the Gaza Strip.
"We will continue with further activities, covert and overt, to ensure Israel's security".
During that war, the armed Shia group was able to overwhelm Israel's ground invasion of southern Lebanon and strike military and civilian targets, undermining internal Israeli support for the war and spurring regional support for Hezbollah's military successes against a state army.
Israel released video footage of digging and pile-driving equipment at work in unidentified locations, carrying out what it said were "tactical preparations to expose Hezbollah's offensive cross-border tunnel project".
Later the military specified it had located one such tunnel dug from a home in the area of Kafr Kela in southern Lebanon that crossed into its territory and was working to "neutralise" it.
Why it matters: The last Lebanon war in 2006 was similarly sparked by a confrontation between Israel and the Iran-backed Shiite militia on the border.
It announced the operation on Twitter on Tuesday, saying that a number of areas close to Lebanon's border, including Metula, had been declared a closed military zone.
The operation comes hours after Prime Minister Benjamin Netanyahu held a last- minute trip to Brussels to meet US Secretary of State Mike Pompeo. While the construction has prompted complaints from the Lebanese army, Hezbollah has not responded, an indication it could remain restrained amid the new operation.
In 2012, Israel became aware of Hezbollah's "plan to conquer the Galilee", as it was termed by Hezbollah leader Hassan Nasrallah. It's "a plan that would coordinate massive barrages of hundreds of missiles with border infiltrations carried out by 5,000 specially-trained Hezbollah terrorists", the report said.
"The digging of tunnels constitutes a blatant violation of Israeli sovereignty", IDF Brig.
The Lebanese military issued a statement saying the situation on the Lebanese side of the border is "calm and stable", adding that army units deployed in the area were carrying on with their usual missions in coordination with United Nations peacekeepers.
Netanyahu has spoken of a sensitive security situation in recent days without providing details, particularly after his Defence Minister Avigdor Lieberman resigned over a controversial Gaza ceasefire last month.
His comments were seen by some at the time as an attempt to save his government, with polls showing wide disapproval among the Israeli public of his handling of the Gaza flare-up in November.
Israel has long voiced concern about Iran's growing military presence along its northern border, warning of an Iranian corridor that could assist in the transfer of weapons to Hezbollah. An Israeli military source said the operation might take weeks to complete.