"My assessment is that you risk broader destabilisation and you risk a whole series of negative consequences if the fighting continues," Milley, chairman of the Joint Chiefs of Staff, told reporters shortly before landing in Brussels for talks with NATO allies.

The top U.S. military officer, Army General Mark Milley, warned on Monday of the risk of broader destabilization beyond Gaza without a de-escalation in the fighting between Israel and Palestinian militants.

After more than a week of fighting, Milley noted Israel's actions were in self-defence but also cautioned that the levels of violence were such that "it's in no one's interest to continue fighting."

"My assessment is that you risk broader destabilisation and you risk a whole series of negative consequences if the fighting continues," Milley, chairman of the Joint Chiefs of Staff, told reporters shortly before landing in Brussels for talks with NATO allies.

"So in my view, I think de-escalation is a smart course of action at this point for all parties concerned."

Israel pummelled Gaza with air strikes on Monday and Palestinian militants launched rockets at Israeli cities despite a flurry of U.S. and regional diplomacy that has so far failed to halt the fighting.

Israel's missile attacks on the densely populated Palestinian enclave killed a top Islamic Jihad commander and left a crater in a seven-storey office building that Israel's military said was used by Gaza's Islamist rulers, Hamas.

Shortly after Milley spoke, six shells were fired from Lebanon toward northern Israel but fell short of crossing the border, the Israeli military said. Israel said artillery was fired at "the sources of the launches" in Lebanon.

Israel fought a 2006 war against Hezbollah guerrillas, who have sway in southern Lebanon and advanced rockets. The border has been mostly quiet since then.

U.S. Secretary of State Antony Blinken earlier on Monday said the United States was working "intensively" to end violence between Israel and Palestinian militants.

Still, the fiercest regional hostilities in years show no sign of abating, with cycles of attacks and counterattacks. A senior Israeli official cast doubt on Monday on the possibility of a ceasefire, telling Reuters: "There is nothing on the table."

Milley broadly cautioned about the risk of fallout from the conflict.

"I believe that whatever military objectives are out there need to be balanced against other consequences," he said, without elaborating.

Asked if he was concerned about risk to the U.S.-backed normalization deals between Israel and Arab nations known as the "Abraham Accords," Milley answered indirectly: "It's possible that there will be negative consequences for things like that."

Scenes of devastation in Gaza appear likely to make it harder for Israel to win diplomatic recognition from Saudi Arabia, experts say. But so far, other rich Gulf states that invested in opening ties with Israel have not shown public sign of second thoughts.