Multinational titans like Apple and Amazon have long sought out more favorable tax arrangements with European Union governments. The extent to which the European Commission is able to rein them in could become clearer this year.

Apple Inc., Amazon.com Inc., Fiat Chrysler Automotive N.V., Starbucks Corp. are entangled in disputes with the European Commission over their intercompany pricing deals with member states, and the commission recently announced it was beginning an investigation into Nike inc.

The Commission has already issued...