Mr. Graham, too, contends that American control of the oil fields would “deny Iran and Assad a monetary windfall,” as he put it in a statement last week.

But Mr. Graham has taken the argument a step further, to suggest that Syrian oil could go into American coffers, as Mr. Trump once implied for Iraq. “We can also use some of the revenue from oil sales to pay for our military commitment in Syria,” Mr. Graham added.

Last week, Mr. Trump offered a variation on that idea, saying that “we’ll work something out with the Kurds so that they have some money, they have some cash flow.” He added that he might “get one of our big oil companies to go in and do it properly.”

But energy and security experts say it is unlikely that any American companies would be interested in the enormous risks and limited profits such an arrangement would entail. Even at its peak, Syrian oil production was modest. And any short-term revenue potential is severely limited by logistical challenges posed by infrastructure damaged by war, pipelines that run into unfriendly areas and the unusually low grade of the oil itself.

Talk of monetizing the Syrian oil also diverges from the message of top Trump administration officials, including Mr. Esper, who said last week that the American mission in Syria was unchanged from its original purpose of defeating the Islamic State.

But the president has repeatedly boasted that the militant group has already been defeated. And although ISIS currently controls no territory, and is little threat to the oil reserves, experts warn that it could regenerate.

Framing control of oil as part of the fight against ISIS, however, may provide cover for an action motivated, at least in part, for reasons that analysts say have no basis in domestic or international law.