The United Nations has issued an urgent plea for calm and respect for the agreed ceasefire after Israel pushed its offensive into Lebanon, a move that puts fragile US-Iran peace talks on the brink of collapse.

"We are deeply alarmed by the escalation in military activities across southern Lebanon and beyond," Stephane Dujarric, spokesman for UN Secretary-General Antonio Guterres, said. "We urge all actors to respect the cessation of hostilities and avoid further escalation."

The UN call came amid Israeli government ordered strikes on the southern suburbs of Beirut on Monday, a day after its ground forces reached their furthest point in Lebanon in 26 years and as Hezbollah fired rockets at northern Israel, including the outskirts of the coastal city of Haifa.

A source close to Hezbollah confirmed to AFP news agency that the Iran-backed group would not stop targeting northern Israel. Hezbollah "has not committed to stop attacking" northern Israel, the source reiterated, requesting anonymity and added: "Why stop these attacks that hurt Israel while it is bombarding Lebanon?"

Israel also continued to threaten to bomb Beirut's southern suburbs if the militants kept up attacks, causing further escalation.