President Obama promises that the U.S. troops he’s sending to Iraq won’t get into another shooting war, but a top White House adviser has listed several scenarios that would prompt direct military action against Islamic militants.

Ben Rhodes, a deputy national security adviser to the president, said the U.S. “has a role to play in a number of different ways” that could result in military engagements in Iraq. The government in Baghdad is fighting militants loyal to the Islamic State of Iraq and the Levant (ISIL), and Mr. Obama has sent nearly 500 U.S. troops to protect the U.S. embassy and the airport in Baghdad, and to serve as military advisers.

“I think the threats that we would look to, for instance, would include an evaluation of whether ISIL is posing a threat to U.S. interests that would necessitate our taking action against them, as we have against terrorist organizations in other parts of the region,” Mr. Rhodes said. “I think the security and safety of our personnel would certainly be of profound interest to the United States.”

In addition to those two possibilities for military action, Mr. Rhodes told foreign journalists Tuesday that the Obama administration would consider engaging ISIL forces if the U.S. believes it can make “a positive difference.”

“We have left that door open if we believe it can make a difference, a positive difference, or if we believe that it is in our core interest to do so because we face a counterterrorism threat or a threat to our personnel,” Mr. Rhodes said. “And I’d add keeping that Embassy open and keeping our operations running in Iraq is what facilitates our ability to cooperate with the Iraqi Government and provide them with security assistance and political support.”

Mr. Obama has vowed that U.S. troops will not engage in ground combat in Iraq, less than three years after he withdrew all American fighting forces from that country. But he has left open the option of air strikes, possibly by drones, to help the government in Baghdad beat back the militants in the northern part of the country.

Gen. Martin Dempsey, the chairman of the Joint Chiefs, has told Congress that the U.S. needs a better understanding of the targets before he would recommend any attack.

“It’s not as easy as looking at an iPhone video of a convoy and then immediately striking it,” Gen. Dempsey said. “Until we can actually clarify this intelligence picture, the options will continue to be built and developed and refined, and the intelligence picture made more accurate.”

Mr. Rhodes said the president “has been very clear that there’s not a U.S. military solution that can be imposed on the current dynamic in Iraq.”

“Our assessment teams are on the ground,” he said. “Our joint operation centers that we are establishing with the Iraqis will help support their efforts to coordinate operations against ISIL as well. But those are Iraqi operations, ultimately.”

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