Wells Fargo & Co. is having trouble getting top bankers interested in its open chief executive officer job.

The bank’s board has approached a small group of top candidates, including JPMorgan Chase & Co. consumer banking chief Gordon Smith, PNC Financial Services Group Inc. CEO William Demchak and former U.S. Bancorp chief Richard Davis, according to people familiar with the matter. Messrs. Demchak and Davis took a pass on potentially replacing Timothy Sloan, who resigned in late March, the people said.

Wells...