Earlier, demonstrators gathered around the presidential motorcade as Trump headed to one of his golf courses.
Democratic candidate Joe Biden won the election on 3 November.
Biden won 306 votes in the electoral college, far more than the minimum required to win the presidency (270 votes).
So far, however, Trump has refused to accept defeat. He has started legal appeals in several key states and has alleged widespread wrongdoing, without providing evidence.
The demonstrations began at noon local time (5:00 pm GMT) near the White House.
Alongside Trump supporters among ordinary Americans, the rallies are expected to be attended by neo-Nazis, far-right groups, and conservative activists.
Organizers and journalists with far-right media, and White House officials, expected large numbers of demonstrators to take part.
Leftist groups are planning to stage opposition demonstrations.
Trump continued to reiterate his allegations of election irregularities. In a tweet on Saturday, he questioned the reliability of the counting of ballot papers sent by mail to Georgia voters, saying, “Expose the crime.”
On Friday, election officials said the ballot was “the tightest in American history,” in the most direct response from federal authorities to the president’s allegations.
On Friday, White House spokeswoman Kylie McKinani told Fox News: “President Trump believes that he will remain, President Trump, he will have a second term.”
But all of Trump’s attempts to turn the score around have failed or have faltered so far.