At issue is the extent of Iranian influence inside Iraq. The hawks have maintained that without a sizable U.S. push in the coming year, Iran will come to dominate Iraq. In the Pentagon and the State Department, senior officials do not such see the situation in such dire terms.

“There’s definitely a perception in the White House among certain people — and I mean the National Security Council — that the ­counter-[Islamic State] campaign was essentially making Iraq safe for Iran,” said one senior U.S. official, who added that the view was not one shared by Mattis or Secretary of State Rex Tillerson.

“I don’t think McMaster sees it that way. . . . What the president thinks, I can’t characterize,” the official said.

The same debate extends to the fight against the Islamic State in its far-flung outposts in places such as Afghanistan, Libya and Somalia. For Mattis and McMaster, the battle against the Islamic State remains a global one. To others, such a formulation is too broad.

“Americans are tired of being globalist in nature,” said the senior administration official who has advocated a narrower focus on the Islamic State in Iraq and Syria. “We’re fighting everyone’s battles for them. I think Mattis and McMaster will push the president to be more globalist in his thinking. But that’s not the vision he was elected on.”