A hurricane warning is in effect for the Cayman Islands, where the storm could dump up to 8 inches (20 centimeters) of rain, said the U.S. National Hurricane Center in Miami.
Paloma may make landfall in Cuba over the weekend, and Jamaica was warned to also monitor its progress, the center said.
The Cayman Islands' top elected official, Kurt Tibbetts, urged residents not to wait until the last minute to prepare for the storm.
"History has taught us that we cannot afford complacency during hurricane season, for conditions can rapidly deteriorate," he said.
Grocery stores reported brisk business by noon on Thursday, and some people had started to board up their windows.
Officials toured low-lying regions, concerned that the onslaught of recent rains could lead to quicker flooding.
As a tropical storm earlier Thursday, Paloma dumped rain on the remote Nicaragua-Honduras border.
On Thursday evening, Paloma was located about 165 miles (265 kilometers) south of Grand Cayman and moving north near 12 mph (19 kph). A gradual turn to the northeast is expected late Friday.