Hurricane Rafael pushed into the Gulf of Mexico on Wednesday night after plowing across western Cuba as a Category 3 hurricane with winds so powerful it knocked out the entire country's power grid.

Massive waves lashed at the shores of Havana as sharp winds and rain whipped at the city's historic centre, leaving trees littered on flooded streets on Wednesday evening.

Forecasters warned Rafael could bring "life-threatening" storm surges, winds and flash floods to western swaths of the island after it knocked out power and dumped rain on the Cayman Islands and Jamaica the day before. The extent of the damage was still unclear as of Wednesday night.

The storm was located 90 kilometres west-northwest of Havana on Wednesday. After plowing across the island, it slowed to a Category 2 hurricane. It had maximum sustained winds of 170 km/h and was moving northwest at 20 km/h, according to the U.S. National Hurricane Center.

The centre predicted storm surges in Florida could reach up to more than 90 centimetres in Dry Tortugas and 30 to 60 centimetres in the Lower Florida Keys.

The storm is bad news for Cuba, which is struggling with devastating blackouts while recovering from another hurricane last month that killed at least six people in the eastern part of the island.

Earlier on Wednesday, crews in Havana worked to fortify buildings and clear scraps from seaside areas in anticipation of flooding.

Classes and public transport were suspended on parts of the island and authorities cancelled flights in and out Havana and Varadero.

Meanwhile, thousands of people in the west of the island were relocated as a prevention measure.

Silvia Pérez, a 72-year-old retiree living in a coastal area of Havana was among those scrambling to prepare. As other neighbours moved appliances and other furniture from ground floor homes, Pérez stocked up on water and food.

"This is a night I don't want to sleep through, between the battering air and the trees," Pérez said. "I'm scared for my friends and family."

Forecasters expected the storm to weaken over Cuba before emerging in the southeastern Gulf of Mexico.

The U.S. State Department issued an advisory for Cuba on Tuesday afternoon, offering departure flights to non-essential staff and American citizens, and advising others to "reconsider travel to Cuba" due to the storm's potential impact. 

The storm on Tuesday knocked out power in parts of Jamaica and unleashed flooding and landslides.

The Jamaica Public Service, the island's electricity provider, said in a statement late Tuesday that impassable roads were preventing crews from restoring power in some areas.

Power outages were reported across the Cayman Islands after a direct hit late Tuesday, and schools remained closed on Wednesday.

"While conditions have improved on Grand Cayman, residents are advised to exercise extreme caution on the roads and near coastlines as rough seas and residual flooding risks may persist," the government said in a statement.

Heavy rainfall also was expected to spread north into Florida and nearby areas of the southeast U.S. during the middle to late part of the week.