Heathrow Airport has a "abysmal" record of controlling building costs said the boss of British Airways-owner IAG.
Willie Walsh told the BBC that other companies should be allowed to design and build any new terminals at Heathrow.
Heathrow said it did not believe such a model was appropriate.
The government is due to publish final proposals for a third runway at Heathrow in the next few months. MPs will then vote on those plans.
Mr Walsh called for an end to Heathrow airport's "monopoly" on its terminals.
"Heathrow Airport Limited run an airport - they're not the best at designing or building the facilities," he said.
"We believe that should be left to others who are much better, and who would have a greater focus on cost control."
But Heathrow's chief executive John Holland-Kaye accused Mr Walsh of "a blatant attempt by Mr Walsh to maintain a dominant monopoly for BA at Heathrow and to frustrate the increased airline competition that should result from expansion".
Mr Walsh does not think that existing terminals should be sold off, but rather that competitors should be allowed to submit proposals for developing, designing and building new terminal facilities.
"It's not rocket science, there are plenty of people who have been involved in this type of construction in other areas who would be interested," he said.
"We know because we've been approached by many of them, who believe that they could do this in a much more efficient way than Heathrow."
Mr Walsh's call is the latest salvo in his ongoing war of words with Heathrow, which is ultimately owned by Spanish firm Ferrovial.
IAG, which owns Iberia and Aer Lingus as well as BA, controls just over half the landing slots at Heathrow - Europe's busiest airport.
Mr Walsh long been a vocal critic of the fees imposed on airlines by Heathrow, which stand at about £22 per passenger.
IAG said the proposed expansion of the airport could allow independent companies to build and operate commercial facilities at the airport, including terminals.
"Heathrow's had it too good for too long and the government must confirm the Civil Aviation Authority's powers to introduce this type of competition," Mr Walsh said.
A spokesperson said Heathrow wanted to ensure competition and choice between airlines at the airport: "Expansion will open up opportunities for IAG, easyJet, flybe, Virgin and dozens of international airlines with whom we are working closely to deliver expansion at close to current charges."
Heathrow chief executive John Holland-Kaye is due to give evidence to MPs on the Transport Select Committee on Monday afternoon.
The third runway was due to cost almost £17bn, but Heathrow argued that it could complete the project for £14bn.
Airlines fear that the airport will increase landing charges to help pay for the third runway.
IAG has threatened to call for a price cap on landing charges if they do not fall after the new runway is finished.
"With more passengers and the introduction of internal competition, the airport's charges should go down," Mr Walsh said.
"If they remain at current levels we, along with other airlines, support a price cap to ensure they cannot rise and have written to the Transport Select Committee to highlight this."