Heathrow officials have claimed that the delays at immigration will mean that passengers would have to be held on arriving aircraft rather than in the terminals for safety reasons.
This would cause further chaos at the airport, with no available parking stands, potentially leading to diversions to airports outside the UK.
Heathrow handles more international passengers than any airport in the world and the airport operator BAA is keen to minimise the effect of the strikes.
Civil servants with no experience and little training are being brought in to help minimise disruption, while border agency workers who are not union affiliated will also be asked to help.
Thousands of border agency staff are expected to walkout in Wednesday’s strike over pensions.
It is estimated that almost two million workers could strike next week.
The industrial action is a protest against proposals which would see public sector workers working longer before receiving their pension as well as having to contribute more.
The government has argued that this is a necessary step as people are living longer and the current pension system is a burden on the taxpayer.