As a first step, the Iranian-backed Hezbollah militant group would stop all attacks on Israel and in return Israel would refrain from escalation in Beirut, a US official said on Sunday.
"This would create space for gradual de-escalation and an effective cessation of hostilities," according to the official.
They added that Aoun tried to advance the proposal and secure an agreement.
However, Lebanese Parliament Speaker Nabih Berri, who claimed to "guarantee" Hezbollah's commitment to a ceasefire, placed the burden on Israel to stop "shooting first".
Netanyahu said on Sunday that troops had moved further into Lebanon in the battle against Hezbollah, despite a ceasefire announced more than six weeks ago.
In the latest advance, Israeli troops seized the 900-year-old Beaufort Castle and a strategic ridge in southern Lebanon, the military said earlier on Sunday, a day after one of the heaviest days of Hezbollah fire toward northern Israel since the April ceasefire, prompting school closures and restrictions.
The US official said that the US did not expect Israel to absorb ongoing attacks on its civilians from Hezbollah.
with reuters
