According to Scottish and Southern Electricity Networks, more than 16,300 houses have lost power as a result of the incident.
"We're aware of a widespread power cut affecting many of our customers around the Hayes, Hounslow and surrounding areas," they said.
With a record 83.9 million passengers last year and a jet landing or taking off almost every 45 seconds, Heathrow is one of the busiest airports in the world.
The cause of the fire is currently unknown.