The pilot of flight BA727 from Geneva in Switzerland reported being hit as the Airbus A320 approached the west London hub on Sunday afternoon with 132 passengers and five crew on board.
Scotland Yard said that no one has been arrested and aviation police based at the airport are investigating.
The Air Accidents Investigation Branch also confirmed that it has begun an investigation.
The collision is the latest and most serious in a string of incidents involving drones at Heathrow, with several near misses between flights and unmanned aircraft reported in the last year.
And it raises the issue of regulation and control of drones, especially in sensitive areas like airports.
BA said the aircraft was examined by engineers and cleared to take off for its next flight following the incident.
Steve Landells, flight safety specialist at the British Airline Pilots Association (Balpa), called for greater enforcement and awareness of rules that govern drone flights.
He said: “Frankly it was only a matter of time before we had a drone strike given the huge numbers being flown around by amateurs who don’t understand the risks and the rules.”
The Civil Aviation Authority (CAA) “drone code” says the unmanned craft should not be flown above 400 feet and kept away from planes, helicopters, airports and airfields. Those with cameras fitted should also be kept at least 50 metres from people, vehicles, buildings and other structures.
A report in March by the UK Airprox Board (UKAB) found there were 23 near misses between drones and aircraft in the six months between April and October last year, including two at Heathrow.
On September 22 a Boeing 777 that had just taken off reported a drone narrowly passed down its right hand side. Investigators concluded the drone was at the same height and within 25 metres of the jet. A report was made to police but the drone operator was not traced.
Days later, on September 30, a drone was flown within a similar distance of an Airbus A319 landing at Heathrow. The jet was flying at an altitude of 500 feet and was on the final approach to the west London airport when the drone was spotted.
The Government is also considering technology to restrict where civilian drones can fly amid growing concerns.
The Department for Transport has confirmed it is also talking to manufacturers about introducing so-called geo-fencing technology in their drones.
A CAA spokesman said it was “totally unacceptable” to fly drones close to airports and anyone flouting the rules can face severe penalties including imprisonment.
A Scotland Yard spokeswoman said a pilot on an inbound flight into Heathrow Airport from Geneva “reported to police that he believed a drone had struck the aircraft” at around 12.50pm.
On investigation it transpired an object, “believed to be a drone”, had struck the front of the aircraft.
A BA spokesman said: “Our aircraft landed safely, was fully examined by our engineers and it was cleared to operate its next flight.”