Hezbollah should not make the "wrong decision" of opening a second front against Israel as it battles attacks by Hamas, a senior U.S. defense official warned Monday.

"We are deeply concerned about Hezbollah making the wrong decision and choosing to open a second front to this conflict," the official told journalists.

A senior Hamas official has said only a small number of top commanders inside Gaza knew about the wide-ranging incursion launched into Israel, but that allies like Iran and Hezbollah "will join the battle if Gaza is subjected to a war of annihilation."

Ali Barakeh, a member of Hamas' exiled leadership, spoke to The Associated Press in his Beirut office as Israel bombarded Gaza and vowed a total blockade of the Hamas-ruled territory.

Israeli strikes on south Lebanon killed three Hezbollah members Monday, the Lebanese group said, as tensions surged after Palestinian militants tried to infiltrate into Israel from Lebanon.

The escalation on Israel's northern border with Lebanon comes two days after Hamas militants launched a historic multi-pronged attack on Israel's southern flank from the blockaded Gaza Strip sparking a fierce war with Israel.

Families in several border towns in southern Lebanon have started fleeing north as Israeli shelling continues in the area.

An Associated Press team saw several cars packed with people and belongings departing Monday. “We tried to flee Aita el-Shaab to Rmeish, but they told us everyone has to stay in their area,” a man said as he and his family tried to flee.

The Lebanese government has been promised that Hezbollah will not intervene in the Hamas-Israel war, caretaker Foreign Minister Abdallah Bou Habib said.

The minister told Asharq al-Awsat newspaper, in remarks published Monday, that Hezbollah would only intervene if Israel started the war.

Qatari envoy Jassem bin Fahed Al-Thani is continuing his meetings in Beirut on Lebanon’s presidential file, a media report said.

“The developing situation in the region has become part of the envoy’s discussions,” al-Liwaa newspaper reported on Monday.

Caretaker Prime Minister Najib Mikati stressed Monday that the government's priority is to maintain security in south Lebanon, a day after Hezbollah and Israel exchanged fire.

In an attack it said had been carried out "in solidarity" with Hamas, which launched a surprise assault on Israel the day before, Hezbollah fired Sunday on Israeli positions in the contested Shebaa Farms border area.

Israel has exerted intensive diplomatic efforts to avoid Hezbollah’s involvement in the ongoing war with Hamas, media reports said.

“The ambassadors of world powers informed Lebanese officials of the threats and repercussions that might result from dragging Lebanon into the current conflict, asking them to exert all the necessary pressures on Hezbollah so that it doesn’t take a decision to get involved in the ongoing fighting,” highly informed Lebanese political sources told ad-Diyar newspaper.

Iranian security officials helped plan Hamas’ Saturday surprise attack on Israel and gave the green light for the assault at a meeting in Beirut last Monday, the Wall Street Journal quoted “senior Hamas and Hezbollah members” as saying.

The surprise assault by Hamas against Israel was a meticulously planned offensive that the Palestinian militant group is capable of keeping up, with a risk of even greater escalation, analysts say.

Hamas can count on a deep arsenal of rockets to use against Israel but key questions include how much support it has received from Iran, which has expressed its backing for the offensive, and whether Hezbollah will enter the fray.