Trump, speaking to masked reporters in the White House Rose Garden, suggested that Pence was in quarantine after his press secretary tested positive, though he did not say it outright.
Katie Miller, the spokeswoman for Pence, who heads the White House’s coronavirus task force, tested positive for coronavirus last week along with a personal valet to Trump.
Three members of the task force have since gone into quarantine - infectious disease expert Anthony Fauci, Robert Redfield, director of the Centers for Disease Control and Prevention, and Stephen Hahn, head of the Food and Drug Administration.
A Pence spokesman denied over the weekend that the vice president was also in quarantine but Trump suggested he was when asked on Monday whether he had considered limiting his contact with him.
“I would say that he and I will be talking about that," Trump said at an occasionally testy press conference, AFP reported.
“During this quarantine period, we’ll probably talk,“ Trump said. “I have not seen him since then.”
“We can talk on the phone,“ he said of Pence. “He tested negative, so we have to understand that, but he comes into contact with a lot of people.”
Trump sought to play down any concerns about an outbreak of the virus in the cramped quarters of the West Wing, where the president works out of the Oval Office.
“We want to keep our country running so we have a lot of people coming in and out (of the White House) and many of those people, most of those people, are tested,“ he said.
“Everybody coming into the president’s office gets tested, and I’ve felt no vulnerability whatsoever,“ he said.
Trump, who is desperately seeking to reboot the economy ahead of the November election, said the US was making “tremendous strides” in ramping up testing and the number of virus infections is “coming down very rapidly.”
He said it was essential to get the country reopened.
“The people want our country open,“ the president said.
“Some states could be moving more quickly,“ he said. “People are dying in the lockdown position too.”
Trump abruptly ended the press conference after an exchange with a CBS News reporter who asked him why when Americans were losing their lives did he keep insisting that the US was doing better than other countries when it came to testing.
“Well, they’re losing their lives everywhere in the world,“ Trump said. “And maybe that’s a question you should ask China. Don’t ask me, ask China that question, OK?”
The CBS reporter, Weijia Jiang, who is of Chinese-American origin, than asked Trump “Sir, why are you saying that to me specifically?”
“I’m telling. I’m saying it to anybody who would ask a nasty question like that, “Trump replied before abruptly ending the press conference.
Though it may appear spacious in movies and TV shows, the West Wing is actually rather cramped, making social distancing difficult.
It houses not just the president’s office but also those of some of his closest advisers, the press briefing room and the desks of White House correspondents, who work elbow to elbow.
Kevin Hassett, a senior economic advisor to the president, told CBS over the weekend that it is “scary to go to work.”
“I would be safer sitting at home in my house than going into a West Wing that with even all the testing in the world and the best medical team on Earth, is a relatively crammed place,“ Hassett told another network, CNN.
In a memo issued on Monday, the White House instructed all staff to wear a mask when inside the West Wing, except when at their desks.
Unlike a number of other world leaders, the 73-year-old Trump has not been wearing a mask in public.
He said Monday that Americans have “learned about face masks.”
“The good and the bad, by the way,“ he said.
“It’s not a one-sided thing, believe it or not,“ Trump said without elaborating.
The United States is the country the hardest-hit by the virus with more than 80,000 deaths and 1.3 million infections.