The Public and Commercial Services union PCS.L said its members, including passport officials, would strike on July 26 and then refuse to work any overtime from July 27 to August 20 in a dispute over job cuts and pay.
"The lives of staff have been made intolerable by these cuts and they're at breaking point," said Mark Serwotka, general secretary of the PCS, one of Britain's most militant unions.
"Ministers have known about these issues for a very long time and need to act now to sort out the chaos they have caused."
The government had already condemned the threatened action after the PCS announced on Wednesday its members had voted for action, indicating only 20 percent of staff had taken part in the strike ballot.
"I do not believe it would be right, I do not believe it would be justified," Prime Minister David Cameron said.
Airport passport desks around London have struggled to cope in recent months and some passengers have had to queue for several hours to get through immigration checks at Heathrow Airport because of a shortage of staff.
The Home Office (interior ministry) has already drafted in extra staff to cope with the hundreds of thousands of travellers expected to arrive London during the Games. (Reporting by Michael Holden; editing by Matt Falloon)
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