BRUSSELS—The European Union's executive body is raising pressure to reduce U.S. influence over the Internet's architecture amid what it called weakened confidence in the network's governance after revelations of U.S. surveillance.

The European Commission, the EU's executive arm, will propose on Wednesday the adoption of "concrete and actionable steps" to globalize essential Web functions—including the assignment of so-called top-level domain names, such as ".com" or ".org"—that remain contractually...