"To advance those talks, the United States proposed a clear sequence: Hezbollah must stop all attacks on Israel. In return, Israel would refrain from escalation in Beirut," said the official, who spoke on condition of anonymity, about the conversations between the three leaders.

Netanyahu vowed Sunday to push Israeli forces deeper into Lebanon, with France prompting the United Nations Security Council to hold an emergency meeting Monday over the expanding offensive.

The US official said Aoun attempted to advance the proposal submitted by Washington, but speaker of the Parliament of Lebanon Nabih Berri's "response was evasive and disappointing." Berri is considered an ally of Iran-backed Hezbollah.

Hezbollah for its part says Israel must cease its hostilities first.

"Hezbollah is following Tehran's lead. It clearly has no interest in the welfare of the Lebanese people," the US official said.

"The fastest way to de-escalate and protect civilians on all sides is for Hezbollah to stop firing immediately," the official added.

The US official's remarks come as another meeting between Israel and Lebanon, which do not have diplomatic relations, are scheduled to take place June 2 and 3 in Washington.

Lebanon's health ministry says Israeli attacks have killed more than 3,412 people since early March, with over a million others displaced.