WASHINGTON — The Obama administration is grappling with how to bridge the gap between its increasingly dire assessment of the threat posed by the Islamic State group and the limited, defensive air campaign it has so far undertaken, which military officials acknowledge will not blunt the group’s momentum.

For months, administration officials have been divided about the threat posed by the Islamic State as it seized parts of Syria and advanced on towns in Iraq. Now, amid new intelligence about its growing strength, a consensus is forming that the group presents an unacceptable terrorism risk to the United States and its allies.

At issue is whether President Barack Obama, elected on a platform of ending the Iraq war, will heed calls for a campaign to contain or destroy the Islamic State, an undertaking that could dominate U.S. foreign policy for the remainder of his term.

Defense Secretary Chuck Hagel said the group poses “a threat to the civilized world,” while Senate Intelligence Committee Chairwoman Dianne Feinstein, D-Calif., called the Islamic State a “terrorist army” that must be defeated. But Obama has not used similar language. He has authorized a limited campaign of targeted airstrikes designed to protect refugees and American personnel in the Kurdish region — but not take out the group’s leadership or logistical hubs.

A strategy to destroy the Islamic State would not require large numbers of American ground troops, but it would amount to a significant escalation from the recent air operations, analysts say. It might also require military action in western Syria, where the group has its headquarters in the city of Ar-Raqqah.