The group has been vulnerable, for instance, to airstrikes coordinated with Iraqi and Kurdish ground forces in northern Iraq in the past month, Mr. Olsen said, noting that as a result, “ISIL is losing arms, it’s losing equipment, and it’s losing territory.”

Despite the attention ISIS has received, when American counterterrorism officials review the threats to the United States each day, the terror group is not a top concern. Al Qaeda and its affiliates remain the most immediate focus. That is because ISIS has no ability to attack inside the United States, American and allied security officials say, and it is not clear to intelligence officials that the group even wants to.

In a speech Wednesday morning, Jeh C. Johnson, the Homeland Security secretary, said, “We know of no credible information that ISIL is planning to attack the homeland at present.”

But a chorus of voices demanding tough action to blunt the advances of ISIS — a chorus that has grown louder with the recent release of videos showing the beheadings of American journalists — appears to have had a substantial impact on public opinion. An NBC News/Wall Street Journal poll conducted Sept. 3 to 7 reveals that nearly half of the country thinks the United States is more at risk of a major terrorist attack than it was before the Sept. 11, 2001, attacks.

While ISIS may have long-term aspirations for war with America, the group’s immediate focus is forming an Islamic state under a puritanical version of Sunni Islam.

American officials have said publicly that their greatest fear is that ISIS has inspired radicals in the West. The concern is that jihadists with American or European passports will fight alongside ISIS or other terrorist groups in Syria, then return home trained to carry out an attack of their choosing. It is not clear that airstrikes against ISIS will, at least in the short term, diminish that threat.

Even a limited air campaign could play into an ISIS narrative that American infidels were intervening on behalf of apostate governments in Iraq and Syria. Airstrikes are also risky because the new Shiite-led government in Iraq is unsettled, officials say. Under Prime Minister Nuri Kamal al-Maliki, the government inflamed sectarian tensions, enraging Sunnis who are not natural allies of ISIS. If American airstrikes are seen as supporting the Iraqi government against the Sunnis, bombings could become ISIS recruiting tools.