Saudi officials gave the footage to Riyadh-supporting domestic media, and said it showed a Hizballah member named Abu Saleh al-Libnani sitting in a tent teaching military tactics to Houthi rebels last summer.
More than 90 civilians and soldiers have died in shelling and skirmishes along the Saudi border with Yemen since March 2015, when a Saudi-led coalition launched a military campaign in Yemen against the Houthi rebels.
"If we have an operation to assassinate a military commander on the Saudi border. A special operations unit would enter [Saudi Arabia] and then assassinate, kill or plant a large explosive device," the man identified by Riyadh as Abu Saleh says, in a distinct Lebanese accent.
A man off camera then asks whether it would be a suicide attack, he answers: "It could be a martyrdom operation, we call them martyrdom operations."
The New Arab could not independently verify the content of the video or the identity of the man named as Abu Saleh.
The release of the video comes amid growing tensions between Gulf states and Lebanon because of Hizballah's "hostile" positions.
Saudi Arabia has long accused Hizballah and the kingdom's regional rival, Iran, of funding the rebels in Yemen and providing them with military weapons.
On Tuesday, Riyadh urged Saudis to leave Lebanon "for their safety", after the kingdom halted a $4 billion programme funding military supplies to Beirut.
Qatar and Kuwait followed with similar travel advisories. But the United Arab Emirates went further, banning its nationals from travel to Lebanon and reducing diplomatic representation there.
Announcing the Saudi aid block a week ago, an official said the kingdom had noticed "hostile Lebanese positions resulting from the stranglehold of Hizballah on the state".
Both Iran and Hizballah reject accusations they have provided military aid to the Houthis.