Scotland Yard said the suspects had been stopped by flying squad officers, assisted by a team from the specialist SO19 firearms unit, as they raided a warehouse containing cash and precious metals at 10am.
The raid began when a white Transit van rammed through the shutters of the Swiss Port Cargo warehouse on the outer perimeter of the airport.
A gang of men got out of the van and threatened warehouse staff with at least one firearm and other weapons, including knives and cudgels.
Police said the thieves were targeting a large quantity of precious metal - said in one report to be gold bullion - that had been delivered to the warehouse. They also attempted to force their way into a secure area containing a large quantity of banknotes.
Armed officers, who were lying in wait, used hatton rounds - bullets designed to deflate tyres and open locks with minimal damage - to safely disable the van before challenging the gang.
Two suspects were understood to have escaped after hijacking a white van and forcing the driver to take them away from the airport in the direction of Stanwell. The driver of the van, which was recovered shortly afterwards, was not injured.
It is understood the robbers forced him to drive them a short way from the airport before continuing their escape by other means. A Scotland Yard spokesman said: "We are looking for them."
Detective Superintendent Barry Phillips, of the flying squad, said: "In preventing this attack, we have foiled what would have been one of the biggest robberies ever committed in the UK. We believe that these men were part of an organised criminal enterprise.
"It is clear that a lot of planning and preparation went into the attack, but the Met's flying squad was one step ahead of the team for some time."
The operation, which was codenamed Cartwright, involved more than 100 officers. It was part of Operation Grafton, an initiative launched by the Met's specialist crime directorate in partnership with Thames Valley police, Surrey police, HM customs and excise and other agencies to tackle high-value crime in and around Heathrow airport.
No shots were fired during the operation, and none of the warehouse staff were injured. One police officer suffered minor injuries.
Assistant Commissioner Tarique Ghaffur, the head of the Met's specialist crime directorate, said: "A professional team of armed robbers have now been removed from society. This is an excellent example of the flying squad, supported by armed officers from SO19, ensuring that the security in and around Heathrow is maintained."
Those arrested, who were aged in their early to late 30s, were taken to west London police stations for questioning.
Had they been successful, the thieves would have made off with an even bigger haul than the gang that carried out the Brinks Mat robbery in 1983. In that raid, six robbers broke into the Brinks Mat warehouse at Heathrow and stole £26m of gold bullion.
A Heathrow spokesman said: "We understand the incident was outside the airport's southern perimeter road. It has not affected airport operations."
Security at the airport came under intense scrutiny in 2002 after two heists in quick succession resulted in the theft of almost £8m.
In February that year, £4.6m in dollars and other currencies was stolen from the airport by two robbers.
In what was seen as a copycat robbery the following month, two robbers bypassed supposedly tight checks to hijack a van carrying more than £2m in cash. They were later caught and jailed.
Those incidents, which came as an embarrassment to airport security officials, sparked a security review by the government.
Ministers demanded an urgent report into the raids amid fears that security - supposed to have been increased after the September 11 2001 attacks on the US - was not tough enough.
