Dr Sean Conley said the president had responded “extremely well” to medication and had “remained stable”.
Mr Trump later said he would probably take another Covid test on Friday and hoped to hold a rally over the weekend.
The president earlier pulled out of next Thursday’s TV debate with Democratic rival Joe Biden.
He said he was “not going to waste my time on a virtual debate” after organisers said it would have to take place remotely because Mr Trump had tested positive for coronavirus.
The move sparked a row about how and when further debates would take place.
What’s the latest on Trump’s health?
In a memo released by the White House on Thursday evening, Dr Conley said Mr Trump was displaying no signs “to suggest progression of illness”.
“Saturday will be day 10 since [last] Thursday’s diagnosis, and based on the trajectory of advanced diagnostics the team has been conducting, I fully anticipate the president’s safe return to public engagements at that time,” the memo added.
Earlier, Dr Conley said that if the president’s condition remained the same or improved throughout the weekend and into Monday, “we will all take that final, deep sigh of relief”.
Speaking to Fox News late on Thursday, Mr Trump said he was feeling “really good” and hoped to hold a campaign rally on Saturday evening, possibly in Florida.
How did the debate row unfold?
It began with the Commission on Presidential Debates announcing candidates would take part in the Miami debate on 15 October “from separate remote locations… to protect the health and safety of all involved”.