A spokeswoman for the Department of Health and Human Services said Tuesday that Dr. Hahn was referring to tests being produced by an outside manufacturer, Integrated DNA Technologies, which are being sent to “a variety of academic, health care and commercial labs around the country" in addition to the public health laboratories that are also receiving them.

“The process of getting a test kit out and putting it into production is not something that happens literally overnight, in particular when you’re talking about a million tests,” said Eric Blank, the chief program officer at the Association of Public Health Laboratories, which represents state and local government laboratories nationwide. “It’s a nice thing to say, and it’s a simple thing to say, but the reality is we are a couple of weeks away from being able to deploy a million tests through this process.”

Dr. Hahn repeated the estimate on Tuesday in testimony at a Senate committee hearing. He told Senator Patty Murray, Democrat of Washington, that although “this is a dynamic process,” the federal government was working with private companies to ship about 2,500 test kits to labs by the end of the week. “That should give us the capacity, in the hands of laboratories once they validate, to perform up to a million tests,” he said at the hearing.

The Association of Public Health Laboratories has said that its labs would be able to conduct about 10,000 tests a day when all of its 100 members that can conduct testing are running. Mr. Blank said that labs can run about 100 tests per day. As of Tuesday, he said 54 of those labs were able to do so, with the rest expected to be up and running by the end of the week.

Dr. Hahn and White House officials have been trying to address the lag in testing caused by botched test kits that were rolled out by the Centers for Disease Control and Prevention early last month, and that agency’s enforcement of strict testing criteria.