FRANKFURT— Volkswagen AG on Wednesday said it will launch a massive recall in January of diesel-powered cars that have been affected by the emissions test scandal that has rocked Europe’s largest auto manufacturer.

The recall will begin in the new year once German authorities have approved the company’s plans to fix the cars, a spokesman said, confirming remarks by Volkswagen Chief Executive Matthias Müller in an interview with newspaper Frankfurter Allgemeine Zeitung. The recall is expected to be completed by the end of 2016.

Volkswagen faces hefty costs, fines and damage claims after U.S. environmental regulators said last month that the German auto manufacturer intentionally installed software in some of its cars that allowed the vehicles to perform better on emissions tests than they would on the road.

Mr. Müller told the newspaper that the company “doesn’t need three solutions, but thousands,” since the EA 189 engine was broadly used in several models with country-specific variations. The company presented technical solutions to Germany’s Federal Motor Transport Authority on Wednesday.

A software update would be sufficient for most cars, but other vehicles would require new fuel injection equipment and catalytic converters, Mr. Müller repeated.