WASHINGTON — The Islamic State is expanding beyond its base in Syria and Iraq to establish militant affiliates in Afghanistan, Algeria, Egypt and Libya, American intelligence officials assert, raising the prospect of a new global war on terror.

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, echoed General Stewart’s analysis in testimony before Congress last week.

But it is unclear how effective these affiliates are, or to what extent this is an opportunistic rebranding by some jihadist upstarts hoping to draft new members by playing off the notoriety of the Islamic State.