The US National Hurricane Centre (NHC) said the storm - "Alex" - was 785 miles (1,260km) south-west of the Azores, with winds up to 85 mph (140km/h).
Storm Alex is moving north-east and is expected to hit the Azores on Friday, prompting a hurricane warning.
It 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 Organisation (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.
A sub-tropical storm over the Atlantic has become a hurricane, the first in January in the region since 1938.