The blaze started in the 26-storey Mont Blanc building in posh Kemps Corner area this evening.
A fire brigade official termed it as a "major" fire.
Fourteen fire engines, seven water tankers and four ambulances have been rushed to the spot.
Civic body sources said six fire-fighters sustained injuries during the operation to control the flames.
"People who reside above the 12th floor are trapped in their apartments. They cannot come down as the smoke had engulfed the entire floor," said Viraz, a resident.
The cause of the fire was not yet clear, fire brigade sources said.
Some residents also told reporters that the building was equipped with fire-fighting apparatus.