Storm Alberto is expected to travel across the eastern Gulf of Mexico and approach the northern part of the US Gulf.
The National Hurricane Centre (NHC) said the storm is set to bring heavy rain and flash floods in western Cuba and southern Florida.
The NHC said the island’s rain totals could reach 10 to 15 inches (25-38 centimetres) — and even 25 inches (65 centimetres) in isolated areas.
Florida’s Governor Rick Scott said: “As we continue to monitor Subtropical Storm Alberto’s northward path toward Florida, it is critically important that all Florida counties have every available resource to keep families safe and prepare for the torrential rain and severe flooding this storm will bring.”
He continued: “If any Florida family doesn’t have an emergency preparedness plan, now is the time to act.”
A tornado “or two” could occur over parts of southwestern Florida and Florida Keys.
Residents in areas that are expected to be affected have started to collect sand that they will use to block any encroaching floodwater.
One resident said: “I’m doing this because every time we have a hard rain, it floods at my house.
“We get water from other neighbourhoods, and water can get up to a foot deep in some places.”
A storm surge watch is in effect along the US Gulf coast, from Crystal River to the mouth of the Mississippi River.
Water could reach up to four feet in some areas, the National Hurricane Centre declared.
Alarms were first raised on Thursday when the NHC estimated a 90 per cent chance a subtropical or tropical cyclone would form in the central or eastern parts of the Gulf of Mexico.
On Friday morning the agency, which is based in Miami, announced an area of low pressure in the Caribbean had developed into Subtropical Storm Alberto.
By the following day, the storm was gathering strength as it moved north.
The National Oceanic and Atmospheric Administration (NOAA) does not expect this year’s hurricane season – which starts on June 1st – to be as destructive as 2017.
Last year was of the most active hurricane seasons on records according to the NOAA, after Hurricane Irma destroyed parts of the Caribbean and southeastern US.