Mr. Vilsack said polls that showed Mr. Sanders and Mr. Trump in front were misleading because they did not reflect the geographic distribution of their support. Most of the passion for both men is concentrated around Iowa’s cities, Mr. Vilsack said. That will help them win caucus gatherings near the cities, but it won’t help them across the rest of rural Iowa, he said.

“Trump has the same challenge that Sanders has, I think,” Mr. Vilsack said. “If you have a big turnout in Des Moines, that’s great. But if you lose Plymouth, O’Brien, Cass, Montgomery, Mills Counties, it can overshadow the fact that you had 350 people at Ward 1 in Des Moines. You’ve got to be smart about this.”