Storm Eleanor was about 200 kilometres (125 miles) northeast of the nation at 4:00 am (0000 GMT) and was moving at a speed of 20 kilometres an hour, the weather service said.
"The wind will initially blow from the south-east at a speed of around of around 40 km/h reaching 110 km/h by midday, strengthening to over 120 km/h," it said in a bulletin.
The country's main international airport operator said the facility would close, with national carrier Air Mauritius cancelling flights scheduled for departure on Thursday morning.
Public transport was also halted and schools shut.
Renowned for its spectacular white sandy beaches and turquoise waters, the remote island nation also lies in the pathway of occasional cyclones.
In January, tropical storm Belal killed one person, left thousands without power, flooded buildings and caused traffic chaos.
About a dozen storms or cyclones occur each year in the southwest Indian Ocean during the November-April season.
In February last year, Mauritius was lashed by heavy rains and high winds from Cyclone Freddy, which caused a wave of death and destruction across southeastern Africa including in Malawi, Mozambique and Madagascar.
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