The flight, which was carrying 74 passengers and four crew, was due to travel to Geneva.
"The pilots rejected take-off at low speed due to an engine problem and subsequently evacuated passengers and crew safely," a Swiss spokeswoman told AFP.
The carrier said four passengers "received on-site treatment by the ambulance services for minor injuries".
The plane was a British-made four-jet Avro RJ100 regional airliner, the company added.
Passenger Mike Mompi, 28, who was flying to Switzerland for a skiing holiday, said the plane was revving up on the runway for take-off when the engine blew out, Britain's The Guardian newspaper reported.
"There was a large bang and flames which grew and grew and large chunks of what looked like chunks of red-hot metal started flying up," Mompi was quoted as saying.
"People started freaking out."
He added: "In about 20 seconds we would have been in the air."
The passengers were reportedly evacuated from the plane on emergency slides.
The aircraft, known as a Jumbolino, is specially designed for short take-offs and landings.
But it's not the first time that this model has caused an incident at London City Airport.
In February 2009, a British Airways RJ100 from Amsterdam crash-landed when one of its wheels failed.
And starting next year, according to the Swiss website, its Avro fleet will be progressively replaced by Bombardier "C" series planes.
The runway at London's City Airport, which predominantly serves business routes in Europe, was closed for about an hour while the aircraft was cleared, an airport spokeswoman said.
Firefighters were called out to deal with a "possible fire" shortly after 3pm but were later stood down, a London Fire Brigade spokeswoman told AFP.
Britain's Air Accident Investigation Bureau said they had deployed a team to "investigate a serious incident" at the airport.
Here's a YouTube video of a Swiss Avro RJ100 "Jumbolino" taking off from Bern's airport: