A U. S. Air Force F-15 fighter jet takes off from the Incirlik Air Base in Turkey. (AP)

President Obama’s order to intensify air attacks in Syria has led to new internal debate over whether to loosen tight restrictions on strikes against Islamic State targets that risk civilian casualties, according to senior administration officials.

But so far, at least, the White House has resisted proposals to change the rules of engagement for the bombing. Each strike, whether against a pre-planned target or one chosen on a “dynamic” basis by patrolling aircraft, is weighed against likely collateral damage and must be individually approved by top officers at the coalition operations center in Baghdad.

“We are trying to develop intelligence to get targets, to leverage opportunities . . . to create strikes that have a strategic effect,” a U.S. military official said. “But we’re going to keep doing those the same way we have done. We will not willy-nilly go after a target because it’s right there, right now.”

That caution has drawn sharp criticism from some Republican opponents of Obama’s strategy, with calls to “carpet-bomb” the Islamic State and place a lower priority on avoiding civilian deaths.

France and Britain, both newcomers to the Syrian air campaign after last month’s Islamic State attacks in Paris, have chafed at the tight guidelines, officials acknowledged. “War is a messy business; you cannot eliminate all risk,” British Defense Secretary Michael Fallon warned in a Sunday Times interview this month.

In Iraq, local commanders have publicly complained about U.S. reluctance to strike targets in populated areas.

According to new figures released by the military Tuesday, about 56 percent of all coalition aircraft return from “strike missions” without having used their weapons, either because of bad weather or a judgment that the risk of civilian casualties is too high.

That compares with estimates as high as 75 percent just a few months ago. But officials attribute the change to improved intelligence that has revealed more targets and allows more precise aim, not any change in the rules. But success in striking the militants has also worked in the opposite direction; they respond by positioning themselves behind and among civilians in urban areas and are less exposed.

Complaints from allies and inside the government, the military official said, are in part “a misperception that . . . the number of targets is so abundant that we’re saying no” to opportunities.

Obama has said he makes “no apologies for us wanting to do this appropriately and in a way that is consistent with American values.”

“If the suggestion is that we kill tens or hundreds of thousands of innocent Syrians and Iraqis,” he said in an NPR interview broadcast Monday, “that is not who we are, and that would be a strategy that would have an enormous backlash against the United States,” including potentially creating more militants than it eliminates.

What is at issue, however, is not the difference between no civilian casualties or tens or hundreds of thousands, but drawing a line somewhere in between. While much of the push for increased flexibility has come from the State Department rather than the military, a senior administration official said, “nobody’s advocating for completely loose rules.”

“There has been a careful discussion about whether or not there is any room in the rules of engagement to relax certain provisions, to put more pressure on ISIL while still drawing an appropriate line,” the official said, using one of several alternative names for the Islamic State.

“The question is, where is that line?” said the official, who like some other officials spoke about internal deliberations on the condition of anonymity. “This is not a new debate. Anytime you’re in an armed conflict, at some point or another, usually at many points, there is going to be a renewed discussion about rules of engagement.”

Air Force Gen. Paul J. Selva, vice chairman of the Joint Chiefs of Staff, told Congress this month that “I haven’t met a soldier, sailor, airman or Marine who wouldn’t ask for looser [rules] in any active fight.” But after consulting with commanders at all levels in the fight against the Islamic State, he said, “I know of no rules-of-engagement restrictions that have prevented us from striking targets and that prevented our forces from being as effective as they can be.”

According to widespread independent accounts, at least 250,000 people have been killed since Syria’s civil war began in earnest in early 2012, with widely varying numbers dividing them between combatants and civilians. Most assessments attribute the majority of civilian deaths to President Bashar al-Assad’s military air and ground forces. Russian airstrikes on Assad’s behalf, which began in late September, have killed more than 400 civilians, the Britain-based Syrian Observatory for Human Rights reported last month.

This week, Human Rights Watch accused Russia and the Syrian government of using cluster bombs — indiscriminate scattershot munitions — that it said have killed dozens of Syrian civilians in recent attacks.

The independent organization Airwars, which compiles and vets reports from other groups and public sources, lists between 405 and 553 civilians killed by the U.S.-led coalition in 76 separate incidents for which there was “a reasonable level of public reporting . . . from two or more generally credible forces.”

The Pentagon has acknowledged at least two incidents in which it says some civilians were killed by coalition airstrikes, which began in Syria more than 15 months ago. It says about a dozen allegations are still being examined, including an early-December bombing raid in the northeastern Syrian village of al-Khan in which between 26 and 39 civilians were reportedly killed.

Before every proposed strike, military officials said, they calculate the “civilian engagement value,” or CEV. Assessments can be based on where the target is located — in an unpopulated area, or in a city next to a mosque or a playground — and how important it is.

“If it’s one ISIL troop, an ISIL rifleman, and he is surrounded by several civilians, then it would stand to reason that it’s not worth the life of those civilians to kill a single rifleman,” said Col. Steve Warren, the Baghdad-based coalition spokesman. “If it were [Abu Bakr] al-Baghdadi,” the Islamic State leader, “standing next to one or two civilians, it might perhaps get looked at differently.”

International law allows for civilian casualties, even intentional ones, providing an action is within the bounds of distinction and proportionality, a somewhat subjective judgment that the military importance of the target is worth it.

“If the target is high enough that there’s a no-kidding strategic effect, then our calculus is different,” the military official said.

Most targeting decisions are made by the general on duty at the Joint Operations Center in Baghdad. Approval for a target with a potentially high CEV beyond the guidelines can get bucked up to a more senior officer or to the Tampa-based Central Command, where teams including officials responsible for legal, operational or engineering issues can weigh in, or even to Washington in extreme cases.

CEV calculations also include the type of munition used. Each weapon has a “blast radius” — a circle within which death or injury is likely to be caused.

An urban target might be struck at night, rather than in the daytime when more civilians are present; the calculation would be the reverse in a residential area. Prevailing winds and the direction in which an aircraft is traveling can send a blast in a predictable direction. These are called “mitigation” measures to avoid civilian casualties.

Correction: An earlier version of this story incorrectly said that at least 250,000 civilians have been killed since Syria’s civil war began in earnest in early 2012. That figure covers all people who have been killed, not only civilians.