On Friday, Mr. Trump is convening his staff at Camp David to deliberate over the policy, which the defense secretary, Jim Mattis, once promised would be finished by the middle of last month.

Mr. Bannon pressed Mr. Mattis to consider the use of private contractors in a one-on-one meeting at the Pentagon last month. He sharply challenged proposals drawn up by the national security adviser, Lt. Gen. H. R. McMaster, for the United States to keep several thousand troops based in Afghanistan and to work closely with the Afghan central government.

Speaking to reporters this week, Mr. Mattis said the idea of using contractors — which was developed by two outside businessmen, Erik D. Prince and Stephen A. Feinberg, at the behest of Mr. Bannon — was “part of the options being considered.”

Mr. Mattis added, “The president’s open to the advice of the secretary of state, and myself and the director of the C.I.A.” — a peculiar assertion to make, given that those three officials are traditionally the top-ranking members of the president’s national security team.

General McMaster has become Mr. Bannon’s nemesis in the West Wing, the leader of what Mr. Bannon has described to colleagues as the “globalist empire project” — a bipartisan foreign policy consensus that emphasizes active American engagement around the world.

Mr. Bannon flatly rejects that philosophy. During the presidential transition, he was spotted in an airport by a New York Times reporter carrying a copy of “The Best and the Brightest,” David Halberstam’s book about America’s misadventure in Vietnam. “I’m having everyone in the transition read it,” he said.

Once Mr. Trump was in office, Mr. Bannon opposed the missile strike on Syria after President Bashar al-Assad used chemical weapons on his own people. He has expressed doubts about sending more troops to Syria or Iraq. He is skeptical of American military intervention in strife-torn Venezuela, a prospect raised last week by Mr. Trump, who surprised administration officials by speaking of a “military option” there.