Intelligence officials estimate that the group’s fighters number 20,000 to 31,500 in Syria and Iraq. There are less formal pledges of support from “probably at least a couple hundred extremists” in countries such as Jordan, Lebanon, Saudi Arabia, Tunisia and Yemen, according to an American counterterrorism official who spoke on the condition of anonymity to discuss confidential information about the group.

Lt. Gen. Vincent R. Stewart, the director of the Defense Intelligence Agency, said in an assessment this month that the “Islamic State”, also known as ISIS or ISIL, was “beginning to assemble a growing international footprint.” Nicholas Rasmussen, the director of the National Counterterrorism Center, said.

The sudden proliferation of ISIS affiliates and loyalist fighters motivated the White House’s push to give Mr. Obama and his successor new authority to pursue the group wherever its followers emerge — just as he and President George W. Bush hunted Qaeda franchises outside the group’s headquarters, first in Afghanistan and then in Pakistan, for the past decade.

“Factions which were at one time part of Al Qaeda and its affiliates, as well as groups loyal to it or in some ways working in tandem with it, have moved on to what they see as more of a winning group,” said Steven Stalinsky, executive director of the Middle East Media Research Institute in Washington, which monitors Arabic-language news media and websites.

In Afghanistan last week, an American drone strike killed a former Taliban commander, Mullah Abdul Rauf Khadim, who had pledged allegiance to the ISIS and had recently begun recruiting fighters. But that pledge seemed to indicate less a major expansion of the ISIS than a deepening of internal divisions in the Taliban.

There is no indication that the ISIS controls territory in Afghanistan, but it has signaled its interest in Afghanistan and Pakistan, and has reportedly sent envoys there to recruit.

Western officials, especially in southern Europe, fear that the three Libyan “provinces” could evolve into bases for ISIS fighters traveling across the Mediterranean, into Egypt or elsewhere in North Africa. Eastern Libya has already become a training ground for terrorists going to Syria or Iraq and a haven for Egyptian fighters staging attacks in the neighboring desert.