Israel's prime minister has ordered attacks on the southern suburbs of Lebanon's capital, Beirut, as the conflict with the Iran-backed armed group Hezbollah continues to escalate.

Benjamin Netanyahu said "terrorist targets" in the Hezbollah stronghold of Dahieh would be struck in response to attacks on Israeli civilians and other violations of a US-brokered ceasefire announced in April that has failed to end the fighting.

A senior Lebanese government official told the BBC that it was relying on US mediation efforts to pressure Israel to end its own violations and prevent further civilian casualties.

On Sunday, Secretary of State Marco Rubio spoke to Netanyahu and Lebanese President Joseph Aoun.

According to a US official, he proposed that, as a first step, Lebanese officials should pressure Hezbollah to stop its attacks on Israel and that, 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.

It came after Israeli troops occupying southern Lebanon crossed the Litani river to seize the 900-year-old Beaufort Castle that sits on top of a strategically important ridge.

Israeli Defence Minister Israel Katz said their campaign was "not over" and that they were "determined to crush Hezbollah's power".

Lebanese Prime Minister Nawaf Salam accused Israel on Saturday of pursuing a "scorched-earth policy and collective punishment".

Lebanon was drawn into the war between the US, Israel and Iran on 2 March, when Hezbollah launched rockets into Israel in retaliation for an Israeli strike that killed Iran's supreme leader. Israel responded with an air campaign across Lebanon and a ground invasion.

At least 3,371 people have been killed in Lebanon since the start of the war, according to the country's health ministry. Its figures do not distinguish between combatants and civilians.

Israel says 24 of its soldiers and four Israeli civilians have been killed over the same period on both sides of the border.