The PCS union has announced its members will strike on July 26, the day before the start of the London 2012 Olympic Games.
The industrial action will involve UK Border Agency staff, which could potentially affect visitors and athletes arriving for the Games.
In a statement PCS general secretary Mark Serwotka said: "The lives of staff have been made intolerable by these cuts and they're at breaking point.
"Ministers have known about these issues for a very long time and need to act now to sort out the chaos they have caused.
"They're acting recklessly in cutting so many jobs and privatising services, and are provocatively refusing to talk to us with a genuine desire to reach an agreement."
The union also warned that "If ministers continue to refuse to negotiate an agreement, the union will announce further action at a later date".
According to the BBC website, PCS said that 57.2 per cent of those who voted had backed strike action, with a turnout of 20 per cent.
Report by Mark Caswell