Pence told reporters earlier Saturday that he and his wife would be tested after his spokeswoman confirmed that a staff member in his office tested positive for the virus, and that the individual did not have close contact with either Pence or President Donald Trump.
Trump also told reporters he had been tested a second time that morning before the briefing.
"I feel great," he said.
Pence said the staffer had mild cold-like symptoms but is recovering well.
At the briefing officials also said they had increased testing capacity but encouraged individuals not to seek testing unless they are symptomatic, saying while the number of available testing kits has improved, the testing process involves the use of protective medical equipment whose supplies are rapidly dwindling.
Trump said the private sector is "stepping up" to address equipment shortages and named Hanes and General Motors as manufacturers that are working to produce more medical equipment.
He also said that while he has signed the War Powers Act that allows him to use the private sector to make up for the shortage in medical supplies, he has not yet invoked it.
The briefing came as the U.S. death toll hit 260 and the number of cases neared 20,000, according to figures from Johns Hopkins University.