One of those doctors said Trump had told them "'I feel like I could walk out of here today.'"
The source, who asked not to be identified, said the next 48 hours will be critical in terms of Trump's care.
Trump left the White House and was moved to Walter Reed National Military Medical Center near Washington on Friday just hours after he was diagnosed with Covid-19.
White House doctor Sean P. Conley told reporters outside the hospital on Saturday that Trump had not experienced difficulty breathing, and currently was not on supplemental oxygen.
"The team and I are extremely happy with the progress the president has made," Conley said.
Conley would not provide a timetable for Trump's possible release from the hospital.
The White House has said Trump, 74, will work in a special suite at the hospital for the next few days as a precautionary measure. He had no public events scheduled on Saturday.
The diagnosis was the latest setback for the Republican president, who is trailing Democratic rival Joe Biden in opinion polls ahead of the Nov. 3 presidential election.
Trump has downplayed the threat of the coronavirus pandemic from the outset, even as the disease has killed more than 200,000 Americans and hammered the US economy.
Conley said Trump had received a first dose of a five-day course of Remdesivir, an intravenous antiviral drug sold by Gilead Sciences Inc that has been shown to shorten hospital stays. He is also taking an experimental treatment, Regeneron's REGN-COV2, one of several experimental Covid-19 drugs known as monoclonal antibodies, as well as zinc, Vitamin D, famotidine, melatonin and aspirin, Conley has said.
Trump announced on Twitter early on Friday that he and the first lady, Melania Trump, had contracted the virus.
He is at high risk because of his age and weight. He has remained in apparent good health during his time in office but is not known to exercise regularly or to follow a healthy diet.
A number of other prominent Republicans also tested positive on Friday, including former White House senior adviser Kellyanne Conway and Republican Senators Mike Lee and Thom Tillis.
On Saturday, a third senator was diagnosed with Covid-19: Republican Ron Johnson, chairman of the Senate Homeland Security Committee. Former New Jersey governor Chris Christie also said he tested positive.
Vice President Mike Pence, who would take over presidential duties if Trump became severely ill, tested negative, a spokesman said. The former Indiana governor, 61, is working from his own residence about three miles from the White House.
British Prime Minister Boris Johnson, who was himself hospitalized with Covid-19 in April, told reporters on Saturday he had no doubt Trump would make a strong recovery.
"He's a naturally obviously very resilient character and I'm sure he'll come through it very well," Johnson said.
Chinese President Xi Jinping sent a message to Trump and his wife on Saturday, wishing them a speedy recovery, Chinese state TV said.
The Trump campaign on Friday suspended in-person campaign events involving the president or his family. But Pence will continue stumping for the president and Trump's campaign said on Saturday that Pence will host an Oct. 8 event in Peoria, Arizona.
Trump's campaign manager, Bill Stepien, also tested positive for Covid-19 on Friday and will work from home, according to a senior campaign official.
Biden pulled ads attacking Trump off the air but otherwise continued his campaign, traveling to Michigan on Friday after testing negative for the virus.
In a Twitter post on Saturday, Biden urged Americans to don masks. "Don't just do it for yourself. Do it for the people you love."
Pence's Oct. 7 debate with Democratic vice presidential candidate Kamala Harris will go forward as planned, organizers said.
The virus could complicate Trump's push to install conservative judge Amy Coney Barrett on the Supreme Court.
Lee and Tillis are both members of the Republican-controlled Senate Judiciary Committee, which is scheduled to begin hearings on Barrett's nomination on Oct. 12.
On Saturday, US Senate Majority Leader Mitch McConnell said all Senate floor activity would be rescheduled until after Oct. 19, but committee work, including Barrett's confirmation hearing, would continue.
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