The flight from Geneva, with 132 passengers and five crew on board, was hit as it approached the London airport at 12.50 p.m. on Sunday, BBC reported.
After safely landing the plane, the pilot reported an object -- believed to have been a drone -- had struck the front of the Airbus A320.
British Airways said it would give the police "every assistance with their investigation".
However, no arrests have been made so far, police said.
The Metropolitan Police's aviation security unit based at Heathrow is leading the investigation.
A Civil Aviation Authority (CAA) spokesman said it was "totally unacceptable" to fly drones -- which are generally used to capture aerial film or photographs -- close to airports.
Steve Landells, from the British Airline Pilots Association (Balpa), said it had been "only a matter of time before we had a drone strike" and called for greater enforcement of existing rules.
In Britain, flying a drone near an airport is a punishable offence with up to five years in prison, and rules also forbid flying them "beyond the direct unaided line of sight" of the operator or near buildings and crowds of people.
Drones over 7 kg cannot be flown above 400ft.