Problems at Wells Fargo & Co., which continues to grapple with the fallout from its sales-practices scandal, have extended to its investment bank, prompting the firm to fire with cause four foreign-exchange bankers, according to people familiar with the matter.

The bank has launched an internal investigation and regulators are also probing the foreign-exchange business, the people said. The issue that led to the firings, along with the reassignment of a senior executive, couldn’t be fully determined. It involves one specific...