Mr Trump tweeted “going well, I think!” after his arrival. US media say his symptoms include a low-grade fever.
He has so far been treated with an experimental drug cocktail injection and the antiviral medication remdesivir after both he and First Lady Melania Trump tested positive for Covid-19.
In exactly one month, Mr Trump faces Joe Biden in the presidential election.
The president’s diagnosis has upended his campaign and also cast doubt on his attempt to get a new Supreme Court judge confirmed before polling day.
The latest update from Mr Trump’s physician, Sean Conley, in a memorandum late on Friday, read: “I am happy to report the president is doing very well.” He said the president was not in need of supplemental oxygen.
Mr Trump was taken to hospital “out of an abundance of caution” with “mild symptoms” and would be there for the “next few days”, the White House said.
The list of other people to have tested positive around Mr Trump include close aide Hope Hicks – believed to be the first to show symptoms – campaign manager Bill Stepien and former White House counsellor Kellyanne Conway. Republican Senators Mike Lee and Thom Tillis have also tested positive.
Mr Trump remains in charge. Vice-President Mike Pence, to whom under the constitution the president would transfer power temporarily should he become too ill to carry out his duties, tested negative.