Two prominent factions within the Republican foreign policy establishment are fighting for control over who will help guide former Florida Gov. Jeb Bush’s policy positions as he gears up for a campaign for the presidency.

In the most recent dustup, Mr. Bush’s political operation put the brakes on a pick for the top foreign policy job in the yet-to-be-announced campaign after worries that the choice would cause too much tension with the neoconservative wing of the party, which includes many prominent donors.