Mr. Romney’s team, according to advisers, is betting that the tea-party groundswell, while energizing fiscal conservatives, doesn’t mark a fundamental shift in the preferences of GOP voters. His team believes activists animated by fiscal discipline and the size of government could see Mr. Romney as a conservative Mr. Fix-It for the economy.

Mr. Gingrich views the tea-party revolt as the dawn of an enduring movement, and he has spent more time courting tea-party activists than forging ties to Republican party leaders. Mr. Gingrich has recast his American Solutions group, launched in 2007 to press for expanded oil drilling and other policy initiatives, into a tea-party clearinghouse of sorts—creating a curriculum to teach activists, small business owners and first-time candidates how to run campaigns. He is keeping in touch with this potential army of volunteers.