Roads around the two-storey office building were closed as it began sliding into a hole where a tunnel collapse brought work on a pounds 300m rail link to a standstill on Friday.
Traffic jams of two miles built up around the airport and passengers had to carry their luggage to and from Terminal 3. Builders worked throughout the night to prevent the building - Camborne House - from collapsing. A spokesman for Heathrow police said: 'The building is at a very weird angle and will probably have to be demolished.'
A Heathrow spokeswoman said flights had not been affected 'The building is listing and has cracks in the walls,' she said. 'However, we now think the building is stable.'
The latest collapse occurred after torrential rain at the weekend. Terminal 3 car park was closed for safety checks and contractors pumped concrete into a tunnel under the building to try to stop it falling down.
Heathrow police said that the building - where the Heathrow Express project team worked until Friday - had been in danger of collapse after engineers were unable to pump concrete in fast enough to stop the land subsiding. Last night it was believed that the building had stopped sliding.
Earth began slipping into a railway tunnel on Friday during construction work on the Heathrow Express Rail Link near the Terminal 3 car park.
Travellers were being advised to leave their cars at home and use the Underground to reach the airport all this week.
The Express Rail Link - which will take just 16 minutes from London's Paddington station to Heathrow - is due to be completed in 1997.
Work on the project, the capital's largest transport infrastructure programme, uses a controversial system, called the New Austrian Tunnelling Method. Investigations into the collapse are likely to concentrate on its use.
Doubts about the method were first raised after a collapse in Munich earlier this year in which two people were killed when a bus fell into a crater in a road.