The US National Hurricane Center said Paloma gathered strength, with sustained winds over 185km an hour, late on Friday as it neared Grand Cayman Island.
Hurricane Katrina was a category three storm when it came ashore near New Orleans in 2005 and swamped the low-lying US city, causing hundreds of deaths. The Cayman Islands, however, are considered less vulnerable to the tropical storms, with structures built to withstand fierce winds and rain.
Paloma was expected to hit Cuba on Saturday, possibly gaining more strength on the way.
Cuba has already suffered $5 billion in damages from two hurricanes this year.
Officials said they had evacuated at least 85,000 boarding school students on Friday and would soon begin moving people from flood-prone areas.
"It's a bad dream," said Ofelia Hernandez, a store worker in Cuba's Camaguey province, where the storm was expected to make landfall.
"Barely two months and once again in checkmate."
As it formed on Thursday, Paloma doused Honduras with heavy rains. The UN estimates 70,000 people have been made homeless there by recent storms.
Hurricane season typically extends from June through the end of November in the Atlantic and Caribbean.