The sudden proliferation of Islamic State 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.

“We don’t want anybody in ISIL to be left with the impression that if they move to some neighboring country, that they will be essentially in a safe haven and not within the range of United States capability,” Josh Earnest, the White House press secretary, said on Wednesday.

The Islamic State began attracting pledges of allegiance from groups and individual fighters after it declared the formation of a caliphate, or religious state, in June 2014. Counterterrorism analysts say it is using Al Qaeda’s franchise structure to expand its geographic reach, but without Al Qaeda’s rigorous, multiyear application process. This could allow its franchises to grow faster, easier and farther.

“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.