Erik D. Prince, a founder of the private security firm Blackwater Worldwide, and Stephen A. Feinberg, a billionaire financier who owns the giant military contractor DynCorp International, have developed proposals to rely on contractors instead of American troops in Afghanistan at the behest of Stephen K. Bannon, Mr. Trump’s chief strategist, and Jared Kushner, his senior adviser and son-in-law, according to people briefed on the conversations.

On Saturday morning, Mr. Bannon sought out Defense Secretary Jim Mattis at the Pentagon to try to get a hearing for their ideas, an American official said. Mr. Mattis listened politely but declined to include the outside strategies in a review of Afghanistan policy that he is leading along with the national security adviser, Lt. Gen. H. R. McMaster.

The highly unusual meeting dramatizes the divide between Mr. Trump’s generals and his political staff over Afghanistan, the lengths to which his aides will go to give their boss more options for dealing with it and the readiness of this White House to turn to business people for help with diplomatic and military problems.