The United States envoy mediating between Lebanon and Israel over their disputed maritime border has met Lebanese President Michel Aoun in Beirut, the president’s office said.

The talks on Tuesday focused on finding a solution to rising tensions between Beirut and Tel Aviv over the boundary between the two countries in the Mediterranean Sea.

Amos Hochstein, a senior adviser for energy security at the US State Department, arrived in Beirut on Monday following an invitation from the Lebanese government.

The invitation came after Israel set up a gas rig at the Karish field, which Israel says is part of its United Nations-recognised exclusive economic zone. Lebanon insists it is in a disputed area.

Israel and Hezbollah, the Lebanese Iran-backed Shia armed group, have been exchanging threats, with Hezbollah leader Hassan Nasrallah last week warning he would strike the gas rig at Karish.

The heavily armed Hezbollah, which has fought several wars with Israel, has repeatedly said in the past that it would use its weapons to protect Lebanon’s economic rights.

Days later, Israel’s army chief Aviv Kochavi threatened Lebanon with “unprecedented bombing” saying that a future war would be very large.

The two countries, which have been officially at war since Israel’s creation in 1948, both claim some 860 square kilometres (330 square miles) of the Mediterranean Sea.

Lebanon hopes to unleash offshore oil and gas production as it grapples with the worst economic crisis in its modern history.

During talks last year, the Lebanese delegation – a mix of army generals and professionals – offered a new map that claims an additional 1,430sq km (550sq miles) as Lebanese territory.

Hochstein did not speak to reporters after his 40-minute meeting with Aoun.

Lebanese media reported before Tuesday’s meeting that Aoun would put forward several proposals, including one which shows readiness to give Israel full control of the Karish field in return for Lebanon getting the Qana field, part of which stretches deep into the disputed area.

During a visit to Lebanon in February, Hochstein handed Lebanese officials a proposal which gives more than half of the disputed area to Lebanon. Lebanon did not respond to the proposal.

Asked what the UN could do to advance the negotiations, Stephane Dujarric, spokesman for the UN secretary-general, said the body can “work with the parties to help them find a solution through dialogues, through discussion, between both sides”.

The dispute over the maritime border is more than 10 years old.

In 2012, Lebanon rejected a US proposal that would have seen the country getting 550sq km (212sq miles), or almost two-thirds of the area, with Israel getting the remaining third.

The offer was known at the time as the “Hoff Line”, named after US diplomat Frederick Hoff who had mediated between the two countries.