He posted a video on Twitter ahead of a second night in hospital, where he is being treated for Covid-19.
The president's physician said late on Saturday that Mr Trump had made "substantial progress" but was "not out of the woods yet".
Saturday had been marked by a flurry of apparently contradictory remarks about the president's condition.
Shortly after the medical team gave a generally upbeat update, White House chief of staff Mark Meadows said the president's condition had been "very concerning".
Sources close to the president said he was not happy about Mr Meadows' comments.
A number of people around the president have tested positive, including First Lady Melania Trump. Many of them attended a crowded White House event last weekend on the nomination of Amy Coney Barrett as a Supreme Court judge. It is being scrutinised as a possible "super-spreader event".
According to US media, the latest positive test has been for Nicholas Luna, a close personal assistant to the president.
The president's positive Covid-19 diagnosis, which he made public in a tweet early on Friday, has upended his election campaign. He faces Democratic challenger Joe Biden on 3 November.
What did Trump say in his video from hospital?
In the four-minute message posted late on Saturday, Mr Trump, dressed in a suit jacket and shirt with no tie, thanked the doctors and nurses at the Walter Reed National Military Medical Center close to Washington DC, where he is being treated.
"I came here, wasn't feeling so well, I'm much better now," he said, later adding: "Over the next period of a few days I guess that's the real test. We'll be seeing what happens over those next couple of days."
He said he wanted to get back on the campaign trail.
"I'll be back. I think I'll be back soon. I look forward to finishing up the campaign the way it was started," he said.
"We're going to beat this coronavirus or whatever you want to call it. And we're going to beat it soundly."
What do we know about his condition?
A series of statements on Saturday painted a confusing picture, but the latest update from Mr Trump's physician, Dr Sean Conley, says "while not yet out of the woods, the team remains cautiously optimistic" about the president's condition.
He could not give a timetable for discharge.
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