The Cuban weather service said the downpours would last until Saturday afternoon, bringing more than 20cm (8in) of rain to some places.
Western and central areas of the country, including Havana, are the worst affected.
Hurricane Agatha hit Mexico last Monday, killing nine people.
Nearly 2,000 people have evacuated their homes in Havana, and around 50,000 people in and around the city are without electricity.
Evacuation centres have been prepared to shelter people from the rain, the state-affiliated Cuban News Agency reported.