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.)