Washington (CNN) Health and Human Services Secretary Alex Azar said Tuesday the department does not know how many Americans have been tested for coronavirus and suggested older Americans avoid large gatherings such as campaign rallies.

"We don't know exactly how many, because hundreds of thousands of our tests have gone out to private labs and hospitals that currently do not report in to (the Centers for Disease Control and Prevention)," Azar told CNN's John Berman on "New Day" when asked how many Americans have been tested for coronavirus at this point. "We're working with the CDC and those partners to get an I.T. reporting system up and running hopefully this week where we would be able to get that data to keep track of how many we're testing."

He continued, "We think we've got through, put at the moment, probably 10,000 a day or could be getting tested by the end of the week, 20,000 a day, according to a study by (American Enterprise Institute) that I heard about. We've got now 2.1 million tests available, and 1.1 million have shipped. We actually have a surplus at the moment that are awaiting orders to be shipped."

Asked by CNN for a copy of the study Azar cited, a spokesperson told CNN that the institute's data on US testing capacity has not been published in a study. Rather, the data was being reported on its "COVID-19 Test Capacity" Twitter account, @COVID2019tests.

The Twitter account tracks testing capacity nationwide from AEI. Resident AEI fellow and former FDA Commissioner Dr. Scott Gottlieb tweeted on Tuesday about the page, which gets updated by his assistant, Abigail Keller, according to the institute. The account was created in July for another project and then "repurposed" for the test capacity tracking, Keller said.