HONG KONG—Tens of thousands of protesters waving U.S. flags and with “The Star-Spangled Banner” blaring from loudspeakers gathered in downtown Hong Kong late Monday to call for U.S. lawmakers to pass legislation that would increase scrutiny of freedoms in the city.

Congress is due soon to consider the Hong Kong Human Rights and Democracy Act, a bipartisan bill that, if passed, would require the secretary of state to certify Hong Kong’s autonomy every year for the city to continue receiving the special treatment that has helped...