A write-up accompanying the survey notes that name recognition is a big role this early in the proceedings, and much has been written about Mr. Romney since he told donors earlier this month he was thinking about a third possible presidential run in 2016.

Sen. Rand Paul of Kentucky and New Jersey Gov. Chris Christie were at 7 percent apiece, and Sen. Marco Rubio of Florida and former Texas Gov. Rick Perry were at 5 percent apiece. Four percent preferred another candidate and 12 percent were undecided.

When voters were asked to choose between Mr. Romney and Mr. Bush in a head-to-head match-up, 49 percent picked Mr. Romney, 32 percent said Mr. Bush and 19 percent were undecided; the two heavyweights met privately in Salt Lake City on Thursday.