Lebanon demands Israel halt offshore gas exploration
The countries, which are still technically at war, last year took part in indirect US-brokered talks to discuss demarcation to clear the way for offshore oil and gas exploration.
But those talks stalled after Lebanon demanded a larger area, including part of the Karish gas field, where Israel has given a Greek firm rights for exploration.
"Lebanon is within its rights to evolve its position according to its interest and as suitable under international law," Aoun told visiting United States envoy David Hale.
Aoun "demanded international experts... draw the line according to international law", the presidency said in a statement.
He also called for a "commitment to not carrying out any oil or gas activities and not starting any exploration in the Karish field and its adjacent waters" until the matter was settled.
The talks last year were supposed to discuss a Lebanese demand for 860 square kilometres (330 square miles) of territory in the disputed maritime area, according to a map sent to the United Nations in 2011.
But Lebanon then said the map was based on erroneous calculations and demanded 1,430 square kilometres (552 square miles) more territory further south, including part of Karish.
Lebanon's outgoing public works minister this week signed a decree to make official Lebanon's demand for the larger area.
Aoun, the caretaker prime minister, and the outgoing defence minister still have to sign it before Lebanon sends it to the UN to make its new demand official.
For his part, Hale on Thursday said the US was ready to continue brokering Israel-Lebanon talks "on the basis on which we initiated these discussions", appearing to reject the Lebanese move towards demanding a larger area.
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