Small-scale operations or "border movements" into Lebanon to take out Hezbollah positions right on the border have begun or are about to, according to two U.S. officials.

Israel does not yet appear to have fully decided whether to launch a ground operation but is prepared for one, the officials said. If a ground operation happens, its scope will likely be limited, sources said. The key is fulfilling the promise to Israelis that the tens of thousands displaced from northern Israel will be able to go home. To do that, decapitating Hezbollah is not enough, the officials said.

U.S. officials received a "few minutes notice" at best, before Hassan Nasrallah's assassination on Friday, the officials said.

In response to reports that Defense Secretary Lloyd Austin blew up when Israeli Defense Minister Yoav Gallant informed him about Nasrallah, the officials said, "We do bear a lot of risks" in the region, and their people were not given a lot of notice to prepare for possible hostile responses to the assassination, which did not please U.S. officials.

One of the officials noted that there is tolerance right now and that U.S. "interests in the region could be badly harmed" if U.S. officials lose their contacts with the Israelis.

The sources said Nasrallah, until the very end, had demanded a Hezbollah ceasefire be tied to a Gaza cease-fire.

Diplomacy remains the only viable long-term solution, even if a military campaign sets the stage for it, officials said.

-ABC News' Matt Gutman

President Joe Biden reiterated his message calling for a ceasefire Saturday when pressed by reporters for more reaction from the Israeli strike targeting Hezbollah's leader as he left church in Rehoboth Beach, Delaware.

When asked whether a ground incursion into Lebanon was inevitable now, Biden only said, "It is "time for a cease-fire."

When asked how the U.S. will respond to attacks on the U.S. warships in the Red Sea, Biden only said "We’re responding."

-ABC News' Michelle Stoddart

Israel Prime Minister Benjamin Netanyahu said that the assassination of Hezbollah leader Hassan Nasrallah "settled accounts" for those who had a hand in killing.

"The elimination of Nasrallah is a necessary condition for achieving the goals we have set: Returning the residents of the north safely to their homes, and changing the balance of power in the region over the years, because as long as Nasrallah lives, he would quickly restore the capabilities we took away from Hezbollah," Netanyahu said in a video message Saturday.

Netanyahu said he came to the decision that what Israel had done to Hezbollah was "not enough" earlier this week when he gave the order to assassinate Nasrallah. But, he warned that Israel's work is "not complete."

"The work is not yet complete. In the coming days we will face significant challenges ... There is no place in Iran or the Middle East that the long arm of Israel will not reach, and today you already know how true this is," Netanyahu said.

"We are determined to continue to strike at our enemies, return our residents to their homes, and return all our abductees," Netanyahu said.

-ABC News' Jordana Miller

The U.S. State Department has ordered the departure of some embassy employees and families in Beirut "due to the volatile and unpredictable security situation."

"U.S. Embassy Beirut personnel are restricted from personal travel without advance permission. Additional travel restrictions may be imposed on U.S. personnel under Chief of Mission security responsibility, with little to no notice due to increased security issues or threats," according to the State Department advisory.

U.S. citizens in Lebanon are also being told to leave the country and the travel advisory for Lebanon remains at "do not travel."

"Due to the increased volatility following airstrikes within Beirut and the volatile and unpredictable security situation throughout Lebanon, the U.S. Embassy urges U.S. citizens to depart Lebanon while commercial options still remain available. At this time, commercial flights are available, but at reduced capacity. If the security situation worsens, commercial options to depart may become unavailable," the advisory said.

"If you choose to remain in Lebanon, be prepared to shelter in place should the situation deteriorate further. The U.S. Embassy strongly encourages U.S. citizens in Southern Lebanon, near the borders with Syria, and/or in refugee settlements to depart those areas immediately," the advisory said.