Tropical storm Elsa is moving over the Florida Straits toward the southern US state after battering Cuba with drenching rain and strong winds, but left the Caribbean island without major damage.

The US National Hurricane Center reported the storm was now making its way towards the west coast of Florida at 19km/h with "conditions beginning to deteriorate across" the Florida Keys, the archipelago at the state's southern tip.

A hurricane watch has been issued "for portions of the west coast of Florida," the NHC said in its latest advisory at 10am Irish time.

Elsa has left a trail of destruction through the Caribbean, claiming at least three lives, and is expected to gain strength after emerging off the coast of western Cuba yesterday evening.

"Slow strengthening is forecast through tonight, and Elsa could be near hurricane strength before it makes landfall in Florida," the NHC said.

Over 100,000 people were evacuated from coastal or low-lying areas as the storm cut through Cuba, with the nation's meteorological institute Insmet reporting winds of up to 100km/h.

But its forecasted westward path meant Florida appeared set to avoid the worst of the storm, with the southwest coast of the state set to experience a glancing blow rather than the direct hit expected earlier.

In Surfside, on Florida's east coast, workers overnight used explosives in the controlled demolition of the still-standing portion of a collapsed apartment building - a job accelerated for fear Elsa might topple the structure.

But Miami-Dade County Mayor Daniella Levine Cava told CNN that officials were "very hopeful" that, with Elsa's current path, they would not have to pause search-and-rescue efforts after all.

Elsa is expected to approach the Florida Keys sometime today on its journey northward.

Dark clouds over Havana as Tropical Storm Elsa approached

Elsa became the first hurricane of the Atlantic season on Friday before weakening to tropical storm status on Saturday.

Its advent represented the earliest date ever that a fifth named storm - which does not typically arrive before August - has struck the region.