The US National Hurricane Centre (NHC) said the hurricane was likely to hit the islands on Friday.
Residents have been told to expect waves up to 18m (60ft) high and wind gusts up to 160km/h.
In calendar terms, Alex is one of the earliest tropical systems to form in the Atlantic Hurricane Basin since records began.
Meanwhile, another tropical storm, Pali, has formed over the Pacific, similarly rare at this time of year.
Scientists have linked the storms to powerful winds and high sea surface temperatures resulting from an unusually strong El Nino phenomenon this year.
The World Meteorological Organization (WMO) has said the 2015 occurrence of El Nino will be among the three strongest recorded since 1950.
Severe droughts and significant flooding in many parts of the world are being attributed to the phenomenon, which occurs every two to seven years.
El Nino is a naturally occurring weather episode that sees the warm waters of the central Pacific expand eastwards towards North and South America.
More about:
