Tropical storm Rafael was chugging past western Jamaica on Tuesday and was expected to strengthen into a hurricane as it headed toward Cuba.

The storm was located 115 kilometres southwest of Montego Bay, Jamaica, on Tuesday morning. It had maximum sustained winds of 95 km/h and was moving northwest at 23 km/h, according to the U.S. National Hurricane Center in Miami.

The centre said it expected "steady to rapid intensification" over the next 24 to 36 hours. The storm was expected to continue to pass by Jamaica on Tuesday afternoon, be near or over the Cayman Islands on Tuesday night and make landfall in western Cuba on Wednesday.

"Rafael is expected to become a hurricane as it passes near the Cayman Islands with further strengthening before it makes landfall in Cuba," the centre said.

On Tuesday morning, Cuba's civil defence authorities called on social media for Cubans to prepare as soon as possible because when the storm makes landfall, "it's important to stay where you are." The day before, authorities said they had issued an evacuation order for 37,000 people for eastern Cuba, in the province of Guantanamo, due to bad weather.

A hurricane warning was in effect for the Cayman Islands and the Cuban provinces of Pinar del Rio, Artemisa, La Habana, Mayabeque, Matanzas and the Isle of Youth.

A tropical storm warning was in effect for Jamaica and the Cuban provinces of Villa Clara, Cienfuegos, Sancti Spiritus and Ciego de Avila, as well as the lower and middle Florida Keys from Key West to west of the Channel 5 Bridge, and Dry Tortugas.

A tropical storm watch was in effect for the Cuban provinces of Camaguey and Las Tunas.

"The storm is currently situated in an atmospheric and oceanic environment that is quite conducive for strengthening," the hurricane centre said — although an expert there said it was still unclear how much Florida and other coastal parts of the United States would be affected.

Schools, government offices close in Cayman Islands

In the Cayman Islands, officials closed schools and government offices as they urged residents to prepare. Long lines were reported at grocery stores as the storm approached.

Jamaica also closed down as a precaution, with the storm's outer winds battering the island late Monday. "We urge all Jamaicans to remain vigilant," said Minister of Local Government and Community Development Desmond McKenzie.

The storm is bad news for Cuba, which is still struggling to recover from Hurricane Oscar, which battered the island last month after making landfall in the eastern part of the country, killing at least six people. The storm also coincided with a large-scale blackout on the island.

Forecasters warned that Rafael would unleash heavy rains across the western Caribbean that could lead to flooding and mudslides, with totals of seven to 15 centimetres and up to 25 centimetres expected locally in Jamaica and parts of Cuba.