The Metropolitan Police said those ignoring the advice must comply with conditions imposed on the two events, including keeping to separate designated areas and dispersing by 1600 GMT.
The warnings came as fears mounted about a repeat of vandalism and clashes with police seen last weekend, and violence between Black Lives Matter activists and far-right groups.
“We are asking you not to come to London, and let your voices be heard in other ways,” Bas Javid, a Met commander, said in a statement, AFP reported.
“I absolutely understand why people want to make their voices heard... But the government direction is that we remain in a health pandemic and people are asked not to gather in large groups.
“By doing so, you are putting your own safety, and that of your family or friends at risk.”
Britain has seen a wave of protests prompted by the death during a US police arrest of George Floyd, an unarmed African-American, which has triggered outrage around the world.
The majority have been peaceful, but demonstrations in London last weekend latterly turned violent while crowds in Bristol, southwest England, toppled a statue to a 17th century slave trader Edward Colston and threw it into the harbor.
Prime Minister Boris Johnson on Friday said the protests had been “hijacked by extremists” while criticizing the targeting of statues as “absurd and shameful” and also urging people not to rally.
Several central London memorials were boarded up as a precaution ahead of Saturday, including one of World War II leader Winston Churchill which was defaced with the word “racist” last weekend and the Cenotaph war shrine.
Former Conservative lawmaker Nicholas Soames, Churchill’s grandson, said the “very, very small, extremely explosive group of people” responsible for the vandalism were “behaving in an unspeakable and cowardly manner”.
“I find it extraordinary that millions and millions of people all over the world who look up to Britain will be astonished that a statue of Churchill and the Cenotaph, our national war memorial, could have been defaced in this disgusting way,” he told the Daily Telegraph.
“It feels like a society that has lost its compass.”
The main protest set for Saturday by Black Lives Matter was instead held Friday to avoid clashes with far-right groups and self-styled “patriots” who have promised to protect memorials.
But some people are still expected to turn out in central London for the original event, which includes a planned march from Hyde Park to Trafalgar Square.
Those gathering there must keep to the square, while any counter-demonstrators must stay on Whitehall to the south, according to the police conditions.