“One of them wants to re-energize the party from within, and the other one wants to re-energize the party from without,” said Alex Castellanos, a veteran Republican consultant.

Applied to the electoral map, the inside route would most likely mean that Mr. Walker would try to capture a band of Midwestern and Great Lakes states filled with the sort of working-class white voters he reflects. He frequently notes that Republicans have not carried Wisconsin since 1984, a not-so-subtle suggestion that he could. He also would surely eye four other Rust Belt states President Obama carried both times: Iowa, Michigan, Ohio and Pennsylvania.

Mr. Bush would almost certainly set his sights on the increasingly diverse states Mr. Obama carried at least once in his two elections: Colorado, Florida, Nevada, New Mexico, North Carolina and Virginia. The voting-age population in each of those states is at least 25 percent nonwhite, and in some of them substantially higher…

Mr. Bush’s message is to challenge and persuade his own party. He sticks to his support for an immigration overhaul and backs the Common Core education standards, unpopular stances with many conservatives. Facing criticism and trying to convert skeptics rather than bending to the will of the party base, Mr. Bush has said, are a sign of “backbone.”

Mr. Walker, by contrast, aims to reinforce what Republicans believe and reassure them that they are right.