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.