May 31 (Reuters) - U.S. Secretary of State Marco Rubio 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 U.S. official said on Sunday.
The U.S. has proposed that 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, 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 U.S. official said that the U.S. did not expect Israel to absorb ongoing attacks on its civilians from Hezbollah.
(Reporting by Humeyra Pamuk and Shubham Kalia; Editing by Himani Sarkar and Lincoln Feast.)
