The US National Hurricane Center (NHC) said maximum sustained winds were near 95 km per hour with higher gusts.
Local forecasters are warning of intense rain for days with the worst in central and perhaps western Cuba.
According to provincial reports, more than 100,000 people have been evacuated in the potential path of the storm, most to homes of family and friends, but thousands also to government shelters.
Evacuations were underway as far west as Havana for fear that rains over the next few days could cause the collapse of dilapidated buildings.