The US death toll is now over 76,000 and states are beginning to reopen.
Six members of Mr Pence’s team were abruptly taken off his plane, Air Force 2, after it was held on the tarmac outside Washington, DC for over an hour on Friday, as he prepared to travel to Iowa to meet religious leaders.
The staff members had had recent contact with Mrs Miller, according to an unnamed US official cited in the media pool report. The president and vice-president had not.
Mrs Miller is the wife of Trump aide Stephen Miller.
During a meeting with Republicans at the White House, Mr Trump told reporters: “She’s a wonderful young woman, Katie.”
“She tested very good for a long period of time and then all of a sudden today she tested positive.”
When asked about the possibility of an outbreak in the White House, Mr Trump told reporters: “All you can do is take precautions and do the best you can.”
He also said he was correct not to wear a mask while visiting the World War II memorial in Washington for VE Day, because the elderly veterans there with him were “far away”.
“Plus the wind was blowing so hard and such a direction that if the plague ever reached them, I’d be very surprised,” he added.
Giving the impression it’s under control
BBC White House correspondent Tara McKelvey
The president has made it clear he does not like the way that a mask looks on him.
He said last month that he believes wearing a mask would not make a good impression on world leaders and others he meets.
“As I greet presidents, prime ministers, dictators, kings, queens … I don’t see it for myself, I just don’t,” Trump told a journalist.
For some, the president’s refusal to wear a mask is more than a question of style. It is a political statement.
Trump and the other officials at the White House want to give people the impression that the health crisis is under control and that soon the economy will be on back on track.
Earlier on Friday, Mr Trump told Fox and Friends that his valets have already started wearing masks.
A day earlier it emerged that a member of the US Navy who serves the president’s food had tested positive for coronavirus.
The president said that the individual had tested negative four days beforehand.
“This is why testing’s not necessarily the answer,” he said, though he noted staff would now be tested daily. “Testing only goes so far.”
Later, White House Press Secretary Kayleigh McEnany defended the measures taken by the White House to keep officials safe.
“We’ve taken every single precaution to protect the president,” Ms McEnany said.
Mr Trump’s chief of staff, Mark Meadows, also told reporters: “This is the safest place you can come to.”
On Friday, the US Food and Drug Administration (FDA) approved the first saliva test collection kit for home use.
The government agency has been criticised for allowing the market to become flooded with test kits that sometimes show false results.
According to the White House coronavirus task force, the US is testing about 248,000 people per day and has tested around 8.1m people so far.