​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.Also Read: Trump seeks fresh changes to Iran deal; draft includes Hormuz reopening, nuclear curbs: ReportThey ​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.Also Read: Iran says no deal with US unless its rights secured