Israeli airstrikes on the historic city of Tyre in southern Lebanon and surrounding areas killed 11 people Tuesday amid an Israeli military warning for the entire city to evacuate.

Israel's army said its forces killed a gunman who had managed to infiltrate Israeli territory from Lebanon and opened fire on its troops.

The Americans have recently conveyed a set of Israeli conditions to the Lebanese side, which were rejected by Speaker Nabih Berri and Hezbollah, Al-Jadeed TV reported on Tuesday.

U.S. Ambassador to Lebanon Michel Issa revealed Tuesday that Washington will present a written paper to Speaker Nabih Berri regarding the so-called pilot zones in south Lebanon, where the Lebanese Army would take control from Israel's occupying forces.

"He told me that he would then accept the pilot zones," Issa said in an interview with MTV.

Let Hezbollah chief Sheikh Naim Qassem "tell me what he wants. Does he want to fight? Let him go and fight; we've seen the results," President Joseph Aoun has said, according to his visitors.

In remarks to the al-Quds al-Arabi newspaper, the visitors said Aoun dismissed reports of contacts between U.S. President Donald Trump and Hezbollah.

The Christian quarter in south Lebanon's Tyre, the last pocket of the coastal city spared from Israeli threats, was emptied out on Tuesday after an unprecedented warning for the city's residents.

Hezbollah does not object to withdrawing its fighters from the area south of the Litani River according to a "point for point" principle, which means that Israel should also pull out its forces in tandem, sources said.

The Israeli army on Tuesday said its forces killed a gunman who had managed to infiltrate Israeli territory from Lebanon and opened fire on its troops.

The Israeli army's withdrawal from south Lebanon would enable the Lebanese state to "extend its authority, end armed appearances and eliminate any justification for the persistence of weapons other than those of legitimate authorities and their armed forces," President Joseph Aoun said on Tuesday.

Christian religious leaders from the southern port city of Tyre called on the international community and Lebanese officials on Tuesday to act quickly to prevent Israel from attacking the Christian district of the city, as airstrikes on nearby neighborhoods killed eight people and wounded dozens of others.

The Israeli military has issued an evacuation warning for the port city, including the Christian quarter, which has been spared so far.

Israel's security cabinet has decided that any Hezbollah rocket on Israel will be met with an attack on Beirut's southern suburbs without the need for prior political approval, Israel's Channel 14 reported on Tuesday.