Rina had maximum sustained winds of 80 mph -- up from 45 mph in late morning, the National Hurricane Center in Miami said in its 11 p.m. EDT advisory.
Rina -- which was pushing hurricane-force winds outward as far as 15 miles and tropical storm-force gales as far as 115 miles -- is expected to strengthen further in the next two days and could become a major hurricane Tuesday, the center said.
The hurricane was about 200 miles southwest of Grand Cayman Island and 340 miles east-southeast of Chetumal, Mexico, moving to the west-northwest at 3 mph, the forecasters said.
Rina was still far enough out to sea that no coastal watches or warnings were in effect.
"Rina [is] gradually becoming better organized as it moves slowly west-northwestward over the northwest Caribbean Sea," the center said.
The storm is expected to stay on the same path through Tuesday, and then gradually turn toward the northwest Wednesday, the forecasters said.
Rina could produce 2 to 4 inches of rain over the Cayman Islands.
Belize, the east coast of the Yucatan Peninsula and adjacent islands should monitor Rina's progress, the hurricane center said.
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