The fire service added a number of vehicles were currently alight on an upper level of the car park.
About 70 firefighters from Stratford, Poplar, Leystonstone and surrounding fire stations, as well as ten fire engines are at the scene.
A 32-metre (105ft) turntable ladder is being used to help extinguish the fire.
On X, one person wrote: “Parked up in Westfield car park had security run up to us and tell us to leave straight away through fire exits could hear bangs on the way down.”
Another wrote they had heard explosions.