The dispute has broken out after Heathrow demanded an “investment recovery” charge of £570 for each train running along its private railway track as well as an “operational expenditure” fee of £107.

The Sunday Times reports that this could cost Crossrail as much as £42 million a year and drive up ticket prices accordingly.

Insiders believe the demands are being made by the airport as it is concerned about the loss of revenue from its own Connect service which would be replaced by four trains an hour between Heathrow and Paddington.

As both sides wait for a High Court ruling, Crossrail is already making alternative plans including stopping at another nearby station so allowing passengers to get a final train to the airport.

A source told the paper: “They’ve tried to keep it as quiet as possible, but it’s so obvious what’s going on.

“If you as a passenger come out of Heathrow terminals, you’ll only ever see Heathrow Express signs. Heathrow bosses want to make as much money as they can from their railway.

“They don’t want people to find out about [other services].”

A Heathrow spokeswoman said: “We are committed to increasing public transport to Heathrow and look forward to the arrival of Crossrail. We need to ensure that track access charges are fair and are waiting on a ruling from the courts.”

Transport for London declined to comment, with a spokeswoman for the Department for Transport adding: “It would be inappropriate to comment on ongoing legal proceedings.”