US President Joe Biden's administration said on Monday that fears of a wider conflict between Israel and Hezbollah militants were "overblown," even as some Israeli officials suggested that violent retaliation could come soon.

Israel and the United States blamed Hezbollah for an attack on Saturday that killed 12 Syrian Druze children in the occupied Syrian Golan Heights town of Majdal Shams.

Hezbollah has strongly denied any involvement, and on Sunday night, Israel's security cabinet gave Israeli Prime Minister Benjamin Netanyahu and Defense Minister Yoav Galant the mandate to launch a military strike in response.

Senior Israeli officials are using strong language publicly to warn Hezbollah that further fighting could break out at any time.

"We are on the brink of a potential regional or global war. Iran is behind all the aggression in the region and its ambitions are global," Politico quoted Fleur Hassan Nahum, Israel's special envoy for foreign affairs, as saying in a text message. "The killing of these innocent children crossed a red line."

Democratic Rep. Gregory Meeks of New York, a member of the House Foreign Affairs Committee, also warned that the region was teetering between peace and “all-out war,” speaking on CNN on Monday morning.

But National Security Council spokesman John Kirby told reporters on Monday that expectations of an all-out war were "overblown," citing "the conversations we've had."

“We still believe that a diplomatic solution can work,” Kirby added. “That’s where our energies should be focused — not on a military solution, and certainly not on a military solution that directly involves the U.S. military.”

As tensions rise in the region, concerns are growing. Turkish President Recep Tayyip Erdogan threatened on Sunday to send troops into Israel to intervene on behalf of the Palestinians. Germany on Monday called on Iran and other countries to prevent an escalation, and airlines suspended routes and cancelled flights to Beirut on Monday amid fears of an escalation.

But the White House is “trying not to feed into the right-wing Israeli narrative that we are going to war,” Aaron David Miller, a former Middle East peace negotiator, told CNN. “There’s no point in speculating… Escalation is not inevitable.”

Two Israeli officials told Reuters that Israel wants to hurt Hezbollah without igniting a regional war.

However, Hassan Nahoum said, "Hezbollah, backed by Iran, knows what to do if it does not want war." Lebanese Foreign Minister Abdallah Bou Habib said the militants would withdraw if Israel stopped attacking them.

Sen. Ben Cardin (D-Md.), who chairs the Senate Foreign Relations Committee, said the situation between Israel and Hezbollah is “heated” right now. “Every time something like what we saw in the last couple of days happens, it increases the potential for escalation,” he said in an interview. “It’s very concerning. The chances of something like this happening again are very high.”

Sen. Thom Tillis (R-N.C.), who is strongly backed by the pro-Israel lobby group AIPAC, argued that Israel must respond decisively: “Israel needs to take the necessary steps based on the nature of the rocket attack that killed children,” he said in an interview. “We need to support everything they do.”

Whatever Israel's response, "we'll have to wait and see" what Hezbollah's response will be, said Jonathan Lord, director of the Middle East Security Program at the Center for a New American Security.

"Israel is unlikely to initiate a ground invasion of Lebanon with so many of its forces engaged in Gaza," Lord said. "But I will be watching closely for signs of the Gaza operation winding down, forces resting and re-equipping, and those forces repositioning in the north."

"Israel is not there yet, but I am not optimistic about the situation," he added.