The British Airways flight from Geneva, with 132 passengers and five crew on board, was hit as it approached the London airport at 12:50 BST on Sunday.
If confirmed, it’s thought it would be the first such incident in the UK.
BA said it would give the police “every assistance with their investigation”. No arrests have been made, police say.
The Metropolitan Police’s aviation security unit based at Heathrow will lead the investigation.
After safely landing the plane, the pilot reported an object – believed to have been a drone – had struck the front of the Airbus A320.
A BA spokesman said: “Our aircraft landed safely, was fully examined by our engineers and it was cleared to operate its next flight.”
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”. He called for greater enforcement of existing rules.