Passport controls and offices could be badly affected by any action in the run up to the opening ceremony of London 2012 on July 27 after the move by the Public and Commercial Services (PCS) union.
It said 57% of its members had backed a campaign of action in protest at cuts, and leaders will meet today to decide their next move.
The union is locked in a dispute with the Home Office over plans to cut 8500 jobs, a 1% pay rise cap following a two-year wage freeze, and privatisation of services.
General Secretary Mark Serwotka said: "Ministers have known about these issues for a very long time but have chosen not to act. They have acted recklessly and irresponsibly in cutting so many jobs and, in the case of the UK Border Agency, they have simply tried to paper over the cracks by deploying severely undertrained staff at our borders.
"If these issues are not resolved, they threaten to seriously undermine the Home Office's ability to provide vital public services, and we cannot sit back and allow that to happen," he added.
Damian Green, the Immigration Minister, demanded the union call off "this irresponsible proposed strike".
Mr Green added: "Any action that disrupts the Olympics will be completely unacceptable and the public will not support it."
The possibility of industrial action came as David Cameron vowed to "go after" G4S for the costs to the public purse it has racked up by failing to deliver on its security contract for the 2012 games.
"Let's be clear," said the Prime Minister during a visit to Afghanistan, "if G4S don't fulfil their contract, we will go after them for the money to make sure that they help pay for the military personnel that have been brought in."
Yvette Cooper, for Labour, said that as troops and police officers were cancelling their leave to fill in for the private security firm, it would be "completely unacceptable to pay G4S a £57 million management fee".
The Shadow Home Secretary added: "The Government has been too hands-off about this contract. They need to step in now to make sure proper penalties are enforced."
As a report on the Games preparations was published by the Commons Public Accounts Committee, Margaret Hodge, its chairwoman, called on the Home Office to "get a grip on Locog (the Olympic organising body) and G4S urgently".
She said no credible explanation had been given for the "astonishing 12-fold hike in management costs, from £10m to £125m, and G4S still has not been able to deliver".
She said the Home Office would "win no medals for transparency" and that "we still did not know just how many security guards will actually be delivered or whether G4S will face any penalty".
Ms Hodge added: "A big concern is the Department has no intention of producing a single auditable account for the Games, drawing together both the costs within the Public Sector Funding Package of £9.3bn and those outside."
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