Last Friday, the US Environmental Protection Agency (EPA) said VW diesel cars had much higher emissions than tests had suggested and that software in several diesel cars could deceive regulators.
Mr Winterkorn said VW would co-operate fully with the US authorities.
In a video statement on the company's website, he said it had "betrayed the trust" of millions of people.
He added: "Swift and comprehensive clarification has now utmost priority.
"To make it very clear, manipulation at VW must never happen again."
In an earlier statement, VW said provision for the scandal would be made "in the profit and loss statement in the third quarter of the current fiscal year". This would pay for "necessary service measures and other efforts to win back the trust of our customers".
It added: "Due to the ongoing investigations, the amounts estimated may be subject to revaluation.