The remains of a destroyed funeral hall hit by a Saudi airstrike Oct. 8, 2016. (Hani Mohammed/AP)

Future American military assistance to Saudi Arabia will hinge partly on whether the gulf kingdom embraces a U.S.-backed cease-fire with Houthi rebels in Yemen, officials said Thursday, as the Obama administration intensifies efforts to distance itself from a bloody bombing campaign.

“It’s not going to help sustain any support . . . if they don’t accept the unconditional cessation of hostilities that we think is absolutely, urgently needed, now more than ever,” a senior official said, speaking on the condition of anonymity to discuss internal deliberations.

In a sign of tensions with a core Middle Eastern ally, officials issued the blunt warning to Saudi Arabia as the White House begins a top-to-bottom review of military aid for the kingdom, including both a massive, long-standing program of arms sales and more-limited assistance for the extended air war over Yemen.

The review follows a grisly Oct. 8 bombing in the Yemeni capital, Sanaa, which capped a series of apparent Saudi strikes on civilian targets and deepened U.S. concerns about involvement in a war that has killed more than 4,000 noncombatants.

The Oct. 8 strike, which killed more than 100 people at a funeral, appeared to be a final straw for administration critics of Saudi Arabia’s handling of the campaign, which they believe has ranged far beyond its original mission to defend the kingdom’s border from Houthi attack.

“We are telling the Saudis that supporting their territorial integrity, their sovereignty, that’s one thing. But their campaign inside Yemen is something else, particularly if they’re not prepared to accept the unconditional, immediate cessation of hostilities that we’ve called for,” the official said. “That will obviously be a factor in our assessment.”

Officials spoke about the review hours after the Pentagon launched Tomahawk cruise missiles against Houthi targets in Yemen, a response to a series of attacks this week on nearby U.S. ships. The Obama administration has blamed those shore-to-sea missile attacks, which inflicted no damage, on the Houthis; the rebels have denied involvement.

The officials said the exchange of missile fire, while potentially complicating efforts to secure an immediate cease-fire between the two parties, would not affect the deliberations about future support to Saudi Arabia.

They signaled determination to continue a gradual reduction in U.S. military aid to the Saudi air campaign that has taken place in recent months. Since the campaign’s start last year, U.S. tanker planes have conducted more than 1,400 missions, offloading tens of millions of pounds of fuel. U.S. personnel also have advised their Saudi counterparts on targeting rebels and avoiding civilian casualties.

But U.S. support has dropped off since a now-abandoned cease-fire earlier this year, officials say. According to U.S. military statistics, the frequency of U.S. refueling missions, which has fluctuated throughout the campaign, fell in late summer from a peak earlier in 2016. Today, only four U.S. personnel remain at the Saudi command center in Riyadh.

“[We are] distancing ourselves both in terms of what we say but also in terms of what we’ve done,” the official said.

U.S. personnel do not approve Saudi targets beyond providing a “no-strike” list of civilian and off-limits targets, officials say.

A spokesman for the Saudi military was not available for comment.

For administration critics of the Saudi campaign, the Sanaa strike, which Human Rights Watch labeled “an apparent war crime,” provided an opportunity to press the case that the U.S. support for the campaign must be curtailed even further.

Officials said reductions in or changes to arms sales, not just to support to the Yemen campaign, would be under discussion during the high-level review.

“Everything is on the table,” the official said. The United States has already halted new sales of cluster munitions to Saudi Arabia.

Officials are open to exploring additional restrictions, but a number of approved sales are already in the pipeline. No matter what the review concludes, officials said, some assistance will continue for defense of the Saudi border, in keeping with a pledge from President Obama.

While support for change appears to be growing, some officials within the Pentagon and State Department continue to stand up for Saudi Arabia, stressing the severity of the Houthi threat the country faces on its borders. Neither will it be easy for the Obama administration to impose further stress on a relationship already damaged by new legislation exposing Riyadh to lawsuits over the Sept. 11, 2001, attacks in the United States.

According to the State Department, the Saudi government agreed in principle after the Sanaa airstrike to accept a 72-hour cease-fire, if that proposal is also embraced by the Houthis. But previous efforts to broker a lasting cease-fire and peace agreement have faltered.

“We encourage the Yemeni government and Houthi-General People’s Congress factions to agree to a cease-fire immediately and without preconditions,” State Department spokesman John Kirby said.

If a lasting cease-fire can be established, Kirby said, political talks can resume.

Ilan Goldenberg, a senior fellow at the Center for a New American Security, said the Obama administration may be stuck with support for a campaign many officials have disliked from the beginning, but which was seen as necessary to mitigate Saudi opposition to the nuclear deal with Iran.

“It’s disturbing when you see these types of bombings,” he said. “But what are you going to do, walk away from the U.S.-Saudi relationship?”

Dan Lamothe contributed to this report.