One early indication suggests he is leaning toward his father’s more pragmatic and restrained philosophy. The former Florida governor is considering naming Meghan O’Sullivan as his top foreign-policy aide; several people familiar with the deliberations describe her as the front-runner for the post.

In many ways, the 45-year-old Ms. O’Sullivan, who now teaches at Harvard, bridges the two Bush worlds. She served as deputy national security adviser for Iraq and Afghanistan for much of George W. Bush ’s second term and was heavily involved in carrying out his Iraq policies. But she wasn’t among the neoconservative advisers who drove the initial decision to invade Iraq, and she is more closely aligned with—and is being promoted by—the kind of pragmatists who dominated George H.W. Bush ’s presidency.

All candidates running for president in 2016 will be asked to articulate what factors would drive them to send U.S. troops into battle. The question would be more delicate for Mr. Bush, should he run, because he would have to navigate the legacies of his father and his brother, whose public image was marred by the wars in Iraq and Afghanistan.