Consider a recent survey by TargetPoint Consulting, a top Republican polling and data-research firm. Roughly two-thirds of likely Republican primary voters polled said they would consider backing Mr. Carson, the highest level of support for any candidate in the field. Just 16% said they wouldn’t, the lowest such tally. Those results include voters who didn’t know who Mr. Carson is.

By comparison, 54% of GOP primary voters would consider voting for Mr. Trump—a big jump from the 20% who said that in May—but a third said they wouldn’t, twice the level of resistance as for Mr. Carson.

Republicans were about as open to backing Florida Sen. Marco Rubio, former Hewlett-Packard Co. Chief Executive Carly Fiorina and former Florida Gov. Jeb Bush.