“New customers are up one-third over last year,” Mr. Obama said. “And the more who sign up, the stronger the system becomes—and that’s good news for every American who no longer has to worry about being just one illness or accident away from financial hardship.”

The administration has said it expects to have about 10 million people signed up and with paid coverage by the end of 2016—and it will likely meet and even exceed that goal. That’s because people still have until Jan. 31 to sign up, and Friday’s six million figure doesn’t include consumers with coverage this year who will be automatically re-enrolled for 2016, or people who sign up for coverage on exchanges that are run by states. The federal exchange, HealthCare.gov, serves 38 states.

Sign-up trends are still below earlier projections, and the goals set by the architects of the law. When the law passed, more than 20 million people were expected to have coverage through the exchange by 2016, according to initial projections from the Congressional Budget Office. On Friday, the administration said it wasn’t ready to comment on whether the 10 million projection will be revised.