Rafael is blowing maximum sustained winds of 113 kilometres per hour after skirting past western Jamaica around noon, the US National Hurricane Centre said.
Residents in Jamaica hunkered down, with four emergency shelters activated, according to authorities. No deaths or injuries were reported as bursts of heavy rain dumped on the island.
As Rafael chugged toward the Cayman Islands, the British territory was setting up its own preparations and bracing for damage to infrastructure, power lines and water systems, the government said in a statement.
Premier Juliana O'Connor-Connolly told Radio Cayman that schools would remain closed on Wednesday.
Storm Rafael will likely brush near western Cuba or be even stronger when it makes landfall on Wednesday local time, the hurricane centre said, and pass over the Gulf of Mexico by Wednesday evening.
"We cannot rule out the possibility that Rafael could intensify into a major hurricane before reaching Cuba," AccuWeather meteorologist Jon Porter said in a statement.
"That would be Category 3 with maximum sustained winds of 111 to 129 miles per hour (180 to 208km/h)," he explained.
In Cuba, officials quickly worked to collect garbage and clear drains, particularly in capital Havana on the western end of the island, according to state media outlet Granma.
Cuba's energy grid collapsed in October, with recovery complicated by the passage of Hurricane Oscar. Another storm would further batter efforts to bring the lights back on.
After passing Cuba, tropical storm conditions could come to the Florida Keys, the hurricane centre's forecast showed.
"It is too soon to determine what, if any, impacts Rafael could bring to portions of the northern Gulf Coast," the centre said.
On the Gulf, oil and gas producers began cutting output and pulling workers off platforms.
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