WASHINGTON—The Supreme Court announced Friday that it would move quickly to decide whether the Trump administration can ask U.S. residents on the 2020 census whether they are citizens, a case infused with time pressures and immigration politics.

The court’s announcement comes a month after a federal judge in Manhattan ruled Commerce Secretary Wilbur Ross committed “egregious” legal violations when he moved to add the citizenship question to the census. The justices will review that decision, adding another chapter to the court’s...