It was another stroke of bad news for Cuba, which has been struggling with blackouts while recovering from another hurricane two weeks ago that killed at least six people in the eastern part of the island.
The storm was located about 140km east-southeast of the Isle of Youth, and about 260km south-southeast of Havana.
It had maximum sustained winds of 160 km/h and was moving northwest at 22km/h, according to the National Hurricane Center in Miami.
Forecasters warned that Rafael was expected to slam into Cuba on Wednesday after dumping rain on Jamaica and the Cayman Islands on Tuesday.
The centre warned of floods, storm surges and mudslides.
The US State Department issued an advisory for Cuba on Tuesday afternoon, offering departure flights to non-essential staff and US citizens, and advising others to "reconsider travel to Cuba due to the potential impact of Tropical Storm Rafael".
On Tuesday morning, the Cuban Civil Defence called on 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 in far eastern Cuba, in the province of Guantanamo, due to bad weather.
A hurricane warning was in effect on Tuesday for a portion of 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 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.
The warning was lifted in Jamaica after the storm passed by the western coast.
Rafael on Tuesday knocked out power in Jamaica and unleashed flooding and landslides.
The Jamaica Public Service, the island's electricity provider, said in a statement late on Tuesday that impassable roads were preventing crews from restoring power in some areas.
Meanwhile, crews in Little Cayman and Cayman Brac were expected to fan out early on Wednesday to assess damage after a direct hit late Tuesday.
Power outages were reported across the Cayman Islands 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.
Forecasters warned Rafael would unleash heavy rains across the western Caribbean that could lead to flooding and mudslides in parts of Cuba and the Cayman Islands.
Heavy rainfall also was expected to spread north into Florida and nearby areas of the southeast US during the middle to late part of the week.
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