Airport operator BAA has written to carriers suggesting a 50% reduction in passenger numbers on each international flight arriving into Heathrow on November 30.
Delays at Heathrow could last as long as 12 hours and airlines were warned of "mass cancellations" of departing aircraft.
Passengers were warned they could face "significant disruption" at border zones as immigration and passport staff join the 24-walkout in the bitter row over pensions.
Delays at immigration are likely to be so long that passengers may have to be held on arriving aircraft, airport operator BAA told airlines.
"This in turn would quickly create gridlock at the airport with no available aircraft parking stands, mass cancellations of departing aircraft and diversions outside the UK for arriving aircraft," BAA added.
The warning came in a letter to all airlines which fly into Heathrow from BAA's Heathrow chief operating officer Normand Boivin, who said BAA had "reluctantly concluded" that the UK Border Agency would be unable to provide a contingency plan to support normal operations.
Cabinet Office minister Francis Maude said disruption at border controls was "absolutely unavoidable" and that responsibility lay with union leaders.
"The responsibility for disruption caused to the public and the economic damage that follows from that and the jobs that will be lost because of this strike lies with the union leaders who balloted while discussions were still going on and those union leaders who, on the basis of very low turnouts in those ballots, calledstrikes on that basis," he told BBC radio.
A Border Agency spokesman said: "Securing the border is our priority and over the last weeks and months we have considered all options to ensure we are prepared for union action.
"We always aim to minimise any disruption caused by the decision of unions to strike, but travellers could see longer waiting times at some ports and airports."
Unions have criticised the agency after it emerged that volunteers are being sought from across the civil service to cover for striking immigration staff.
Mr Boivin told the airlines that Heathrow would do all it could and "may also be obliged to advise arriving passengers ... that they should avoid arriving into the UK on November 30 unless absolutely necessary".
British Airways and Virgin Atlantic Airways both announced that the normal charges for rebooking flights would not apply for those wishing not to travel on November 30.
Gatwick Airport's chief operating officer Scott Stanley warned that the industrial action was likely to affect arriving flights at airports across the UK, adding: "We recognise that this will be a challenging time for all airport passengers, and we would warn passengers to be prepared for the potential for significant disruption at the border zones on November 30.
"Unfortunately, airport staff cannot man the desks at the border zone but we will provide all necessary assistance to the Border Agency and we are determined to make sure that the needs and welfare of all our passengers will be met on the day.
"Hundreds of airport volunteers are planning to come in to provide the support and assistance needed on the day.
"All non-essential work and meetings have been cancelled and all staff and managers will be out in the terminals providing information, refreshments and advice and to make passengers feel as comfortable as is practicable given the challenges we expect to face. We will keep our passengers informed of developments."
Deputy Prime Minister Nick Clegg said during a visit to Leeds: "Clearly we're going to do everything we can - notwithstanding the terrible disruption and, frankly, cost to the economy these strikes will create - to make sure that systems, including our borders are properly policed and properly controlled.
"But my message to the trade union bosses who are corralling their members into conflict is: let the teachers and the nurses and the doctors, and so on, let everybody take a bit of time to look at the information that we've provided on our Government website so people can look for themselves at what we're offering."
Unison today launched a set of adverts showing the "real face" of the pensions crisis, including a custody detention officer, a nurse and a local government worker who finds jobs and apprenticeships for unemployed young people.
General secretary Dave Prentis said: "These are real people taking real action to protect their pensions.
"The majority of public sector workers are women, who are struggling to pay their bills and feed their families in the face of a pay freeze and rising inflation. These workers cannot afford to pay more and work longer, to receive less in retirement."