The US-based National Hurricane Centre (NHC) warned that Paloma was "an extremely dangerous" hurricane.
It has warned that flash floods and mudslides are likely, as well as storm surges 2.1m to 2.7m above normal tidal levels, with "large and dangerous battering waves".
Paloma is the 16th storm of the Atlantic storm season and the eighth hurricane. It has already brought heavy rain to parts of Honduras and Nicaragua.
Residents of the Caymans, Cuba and Jamaica have all been put on alert.
Donovan Ebanks, chairman of the Caymans' National Hurricane Management Council said it was "time to stay indoors" while acting police commissioner David George said residents should not leave shelter unless it was essential to do so.
Authorities in the Cayman Islands had urged residents not to leave storm preparations to the last minute.
Residents of the Cayman Islands have been told to safeguard their property
"History has taught us that we cannot afford complacency during hurricane season, for conditions can rapidly deteriorate," said the islands' top elected official, Kurt Tibbetts.
In Cuba, the government has issued hurricane warnings for the provinces of Sancti Spiritus, Ciego De Avila, Camaguey and Las Tunas and has begun to prepare for evacuations.
A hurricane watch has also been issued for Granma province.
Officials in every province were working to "guarantee the protection of the public and economic resources", the Civil Defence said.
Paloma would be the fifth hurricane to hit Cuba this season.
Gladys Sanchez, a resident of Minas, north of the central city of Camaguey, told AFP that many people were still homeless after the earlier storms.
"It has been raining here since morning, everything is dark," she said.
Gustav and Ike, which struck Cuba on 30 August and 9 September respectively, caused billions of dollars worth of damage and destroyed hundreds of thousands of homes.
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