"We have not given a firm deadline to Israel, not really our role. This is their conflict. That said, we do have influence, even if we don't have ultimate control over what happens on the ground in Gaza," Finer told the Aspen Security Forum in Washington.

Israel has two objectives in Gaza, Finer said, ensuring Hamas can no longer govern the densely-populated enclave and it can no longer pose a threat to Israel following the deadly Oct. 7 attack during which its militants killed 1,200 people and abducted 240 others.

"Frankly, if the war were to stop today, (Hamas) it would continue to pose (a threat) which is why we are not in a place yet of asking Israel to stop or to force ceasefire," Finer said.

The United States believes that there are many "legitimate military targets" that remain in southern Gaza including "much if not most" of the Hamas leadership, Finer said, citing Israeli information but added that Washington did not have anything to contradict that assessment.

There have been aspects of how Israel's military operations in northern Gaza was conducted that did not show "sufficient care" for civilian life, Finer said, and repeated U.S. calls to ameliorate those aspects.

"We work day in day out and engage directly day in day out, including today, frankly, between the President and the Prime Minister on how the conflict is being conducted in all parts of Gaza but with a particular focus on the south," Finer said.

He was referring to a call that took place earlier on Thursday between U.S. President Joe Biden and Israeli Prime Minister Benjamin Netanyahu. There were no immediate details of the call but a statement is expected later.

Israel unleashed its military campaign in response to a surprise Oct. 7 incursion by Hamas fighters who rampaged through its towns, killing 1,200 people and seizing 240 hostages, according to Israel's tally.

In retaliation against Hamas, Israel is bombarding Gaza, bombarding from the air, imposing a siege and launching a ground assault. Gaza's Health Ministry says that so far more than 17,000 people have been killed in the enclave of 2.3 million.

Top U.S. officials, including Vice President Kamala Harris and Secretary of State Antony Blinken, have urged Israel publicly to conduct a more surgical offensive in southern Gaza to avoid the heavy civilian casualties inflicted by its attacks in the north.

But since a temporary truce between Israel and Hamas broke down last Friday, hundreds of Palestinians have been killed in Israel's offensive in the south.

(This story has been refiled to correct the name of the White House national security aide in paragraph 1)