Holidaymakers were being given advance notice to “change travel plans”, the group added.
It is understood the plans to use drones are contained in a document shared between members of the group.
Heathrow said the plan was “reckless” and could “endanger lives”.
A statement on the XR website said: “Extinction Rebellion demands the Government begins to act on its declaration of a Climate and Environment Emergency by cancelling all Heathrow expansion.
“On June 18, we plan to carry out nonviolent direct action to ensure Heathrow Authorities close the airport for the day, to create a ‘pause’ in recognition of the genocidal impact of high carbon activities, such as flying, upon the natural world.
“If the Government does not cancel all Heathrow expansion, Extinction Rebellion will act to shut the airport down for up to 10 days from July 1.
“Extinction Rebellion is in the consultancy stage with its members on the proposed action.”
A small group of mostly teenage activists briefly unfurled a banner near a tunnel which leads to Terminals 2 and 3 as several police officers watched on.
The statement, published on Friday, added: “This is not about targeting the public, but holding the Government to their duty to take leadership on the climate and ecological emergency.
“The addition of the planned third runway would make Heathrow the single biggest carbon emitter in the UK – to expand the airport at this critical point in history would be madness.
“We understand the action will cause disruption to a great number of holidaymakers, however we believe that it is necessary given the prospect of far greater disruption caused by ecological and societal collapse, if we don’t act now.”
A Heathrow spokesman said: “This is reckless action that if carried out could endanger the lives of the travelling public and our colleagues.
“We agree with the need to act on climate change, but that requires us to work together constructively – not commit serious criminal offences just as hard-working people prepare to spend a well-earned holiday with their family and friends.”