The White House seems to be leaning toward the Islamic State, increasingly alarmed by what Lisa Monaco, President Obama’s homeland security and counterterrorism adviser, recently called the group’s “unique threat” to the United States.

The debate is evolving in real time, thus there have been no large shifts in money or personnel yet in one direction or the other. But it is the first time senior American officials have spoken so openly about the evolution.

How much the United States spends on counterterrorism is difficult to pinpoint because many of the main actors and agencies — American troops, C.I.A. analysts and F.B.I. agents, to name a few — carry out other functions, as well. But senior American officials say that counterterrorism programs employ roughly one in four of the more than 100,000 people who work at the C.I.A. and other intelligence agencies, and account for about one-third of the $50 billion annual intelligence budget.

About 3,400 American troops in Iraq are helping the Iraqis fight the Islamic State, while about 9,800 forces in Afghanistan are assisting that country’s security personnel in combating the Taliban, Al Qaeda and other extremists there.

The issue is likely to gain prominence in the 2016 presidential campaign, as Republican candidates criticize the Obama administration for failing to anticipate the rise of the Islamic State from the ashes of the Iraq war. “We didn’t finish the job,” Senator Marco Rubio of Florida said last month.

The debate was brought to the surface two weeks ago when James B. Comey, the F.B.I. director, said at the Aspen Security Forum in Colorado that the Islamic State posed the greatest danger to the homeland.