US Secretary of State Marco Rubio has spoke with both Lebanese President Joseph Aoun and Israeli Prime Minister Benjamin Netanyahu on the diplomatic negotiations between Israel and Lebanon and has proposed a plan to allow for "gradual de-escalation", a US official said on Sunday.
The US has proposed that as a first step, the Hezbollah would stop all attacks on Israel and in return Israel would refrain from escalation in Beirut, the official said.
"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 had said on Sunday that he ordered troops to move 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.

US, German scientists win Nobel medicine prize for work on light and brain
Nigeria military aircraft crashes with 32 aboard
At least 22 dead after boats collide on Congo river
Spanish PM Sanchez calls snap election for November 29
Hundreds of French schools stay closed ahead of nationwide day of student protests