The Iranian military's joint command said Monday it was halting its offensive operations, after Israel and Iran exchanged fire in their first attacks since the U.S. struck a ceasefire with Tehran two months ago.

The escalation threatened to drag the Middle East back into a full-scale war.

Israeli bombardment has damaged a World Heritage site in the southern city of Tyre, a culture ministry official told AFP on Monday, as correspondents there saw debris and damage.

One of the oldest cities on the Mediterranean coast, Tyre lies around 20 kilometers from the Israeli border, and its UNESCO World Heritage listed ruins are located in two main areas of the city.

U.S. Ambassador Michel Issa met Monday with President Joseph Aoun, as Israel and Iran traded fire after Israel's bombing of Beirut's southern suburb.

Issa said the U.S. places great importance on Lebanon and has decided that Sunday's events must not expand further. "What happened yesterday is a political message, and we in the United States have decided that the confrontation must not expand any further," Issa said after meeting Aoun.

Israeli strikes targeted overnight into Monday Habboush, Srifa, Zefta, Yater, Nabatieh, Safad al-Battikh, Sojod, Jabal Safi, al-Mansouri, Ainqana, Qalaway, al-Sharqiyeh and al-Hallousieh, as Israel and Iran exchanged attacks over Israel's bombing of Dahieh.

Four people were killed in the strike on Zefta, the national News Agency said, adding that several others were killed in Srifa.

The Israeli military has vowed to press ahead with its military campaign in Lebanon and said it would step up operations against Hezbollah.

"The (Iranian) regime is attempting to establish a new equation through direct attacks on Israeli territory in response to IDF operations in Dahieh," military spokesman Brigadier General Effie Defrin said Sunday, in a televised statement.

Iran, Hezbollah, and others have repeatedly dragged both Israel and Lebanon into "pointless and destructive conflicts" and "Hezbollah has a simple choice: it can continue fighting a pointless war, or it can finally allow the return of the displaced and the reconstruction of Lebanon," a U.S. official said on Sunday.

In remark to U.S. news portal Axios, the official claimed "the terms on the table are fair, have the consent of both sovereign governments, and provide a clear path to end the fighting."

The Israeli military said Sunday it uncovered a sprawling tunnel network beneath the Beaufort castle in southern Lebanon, saying it was built to give Hezbollah a fortified strike hub just kilometers from Israeli territory.

Israeli soldiers recaptured the crusader-era castle recently, giving the military a strategic toehold it once occupied for nearly two decades.

Iran threatened to retaliate against the United States and Israel for fresh strikes in Beirut's southern suburbs on Sunday, further dampening hopes for a lasting peace as the Middle East war reached its 100th day.

U.S. President Donald Trump called for "more surgical" strikes against Hezbollah in Lebanon and said he is not demanding the conflict be included in a peace deal with Iran, in an interview broadcast Sunday.

"I'd like to see a more surgical attack on Hezbollah. I think it should be more surgical," Trump told NBC's "Meet the Press," according to a transcript of the interview recorded Friday.

Israel's military struck alleged Hezbollah command centers in Beirut's southern suburbs in response to rocket fire on northern Israel, Prime Minister Benjamin Netanyahu's office said Sunday.

"In accordance with the directives of Prime Minister Netanyahu and Defense Minister Katz, the Israel Defense Forces have just struck a militant command center in Beirut's Dahiyeh district, in response to Hezbollah's fire towards Israeli territory," his office said.