In a letter to the transport secretary, the London councils and Greenpeace said judicial review proceedings will begin if the government fails to comply.
The group said expansion plans were "unlawful" and a "fair and lawful consultation" had not been carried out.
The government said it would "strongly defend any challenge".
The 33-page letter has been sent by Hillingdon, Richmond, Wandsworth and Windsor and Maidenhead Councils, along with Greenpeace and a Hillingdon resident.
The group said they had set out a "comprehensive legal challenge" on the basis that "the government's approach to air quality and noise is unlawful".
They also said a correct consultation exercise had not been carried out by the government "prior to issuing its decision".
But a spokesperson for the Department for Transport said expansion "can be delivered in line with our obligations on air quality".
The spokesperson said: "We are confident that our process for decision-making is robust and will strongly defend any challenge."
Sadiq Khan, the mayor of London, has directed Transport for London (TfL) to provide "expert advice and assistance" to the councils for  any court proceedings.
Four councils have threatened to begin legal action if the government does not withdraw its support for Heathrow Airport expansion within 14 days.