Volkswagen AG is poised to name Porsche chief Matthias Müller to run the entire company and is demanding resignations from several of the company’s top engineers to confront an emissions-cheating scandal that has spread from the U.S. to Europe.

The moves come a day after Chief Executive Martin Winterkornresigned over a crisis that has cost the company nearly a third of its market value. Mr. Müller’s promotion is expected to be appointed at a meeting of Volkswagen’s supervisory board on Friday, people familiar with the situation...