The dispute escalated in early June after Israel moved a production vessel near the Karish offshore field, which is partly claimed by neighbour Lebanon.

This prompted Beirut to call for the resumption of US-mediated negotiations on the demarcation dispute.

"I remain optimistic that we can make continuous progress as we have over the last several weeks and I look forward to coming back to the region and being able to make the final arrangements," Amos Hochstein told reporters after meeting Lebanon's top leaders.

Hochstein, on his second visit in less than two months, is carrying an Israeli proposal in response to a demarcation offer made by Lebanon.

On Monday, he met with President Michel Aoun, Prime Minister Najib Mikati and Parliament Speaker Nabih Berri at the presidential palace.

Lebanon is looking to clinch "a deal that preserves its rights and its wealth and that could provide, as soon as negotiations are over, an opportunity to revitalise the economy," Aoun said before Monday's meeting.

Lebanon and Israel have no diplomatic relations and are separated by a UN-patrolled border.

They had resumed maritime border negotiations in 2020 but the process was stalled by Beirut's claim that the map used by the United Nations in the talks needed modifying.

Lebanon initially demanded 860 square kilometres (330 square miles) in the disputed maritime area but then asked for an additional 1,430 square kilometres, including part of the Karish field.

Israel claims that the field lies in its waters and is not part of the disputed area subject to ongoing negotiations.

- 'Pay dearly' -

A Lebanese official in mid-June said Beirut had made a new offer to Hochstein, holding back on demands for territory where Israel planned to imminently extract gas.

Beirut was pushing for the country's maritime border to exclude Karish and include the whole of a nearby field instead, the official told AFP at the time.

An Israeli official on Sunday said Israel's offer, which related to the original disputed zones, was a "compromise to both sides" and would allow Lebanon to develop "the Sidon reservoir", known as the Qana field, while safeguarding Israel's interest.

On July 2, Israel said it had downed three drones launched by Lebanon's Iran-backed Hezbollah that were headed towards the Karish gas field.

The Iran-backed Shiite Muslim movement on Sunday released a short video it said showed surveillance of several Israeli-chartered energy infrastructure ships, including the production vessel sent to Karish which is operated by London-listed firm Energean.

Hezbollah's chief Hassan Nasrallah had threatened attacks if Israel proceeds with gas exploration in the disputed area.

On Sunday, he said he was trying to "strengthen" Lebanon's negotiating power and said that his party will determine its next step based on the outcomes of Monday's talks.

"We're paying no attention to Nasrallah's threats and they're not a factor in the negotiations," Ram Ben Barak, head of the Israeli foreign and defence committee told public broadcaster Kan radio on Monday.

"But if Nasrallah dares to do anything to Israel's gas rigs, Lebanon -- and Hezbollah -- will pay dearly."

Israel and Hezbollah last fought a devastating war in 2006.