Lindsey Graham and Rick Santorum want to deploy 10,000 American troops in Iraq as part of a coalition with Arab nations against Islamic State militants, and will settle for nothing less than “destroying the caliphate,” in Mr. Graham’s words.

Jeb Bush believes those additional American soldiers would have prevented the Islamic State from gathering strength in recent years. But an American-led force now? “I don’t think that will work,” he said in an interview Friday, his latest sign of wariness at the prospect of becoming the third President Bush to dispatch ground troops to the Middle East.

Marco Rubio describes his strategy against the Islamic State with a line from the action movie “Taken” — “we will look for you, we will find you, and we will kill you” — yet he is more inclined to provide “the most devastating air support possible” rather than send in American troops. Scott Walker and Rick Perry are more open to a combat mission, while Rand Paul wants boots on the ground — as long as they are “Arab boots on the ground.”

As President Obama grapples with the unnerving territorial gains of the Islamic State last week, the Republicans eyeing the White House are struggling to put forward strategies of their own. The most detailed ideas have come from Mr. Graham, a United States senator from South Carolina who is on the Armed Services Committee, yet he ranks so low in polls that it is unclear if he will qualify to participate in the coming candidate debates. Mr. Bush, a former governor of Florida, and Mr. Walker, the governor of Wisconsin, draw more support from voters at this point, yet seem less sure of their war footing, saying they would rely on guidance from military advisers.