However, only 20 percent of the 16,000-member Public and Commercial Services Union (PCS) even bothered to participate in the ballot -- 57 percent of them voted to go on strike.
PCS said the industrial action is due to worries over job cuts, pay and privatization.
The lives of staff have been made intolerable by these cuts and they're at breaking point,” said PCS general secretary Mark Serwotka.
“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 privatizing services, and are provocatively refusing to talk to us with a genuine desire to reach an agreement.
The transport workers of East Midlands Trains will also strike at the same time.
A separate union, the Immigration Service Union, which represent other border staff, will work as normal.
Government and business officials have condemned the strike, which will occur when Heathrow Airport will be expecting in excess of 100,000 arrivals per day.
Theresa May, the Home Secretary, blasted the union.
“I think that is shameful frankly,” she said. “They are holding a strike on what is one of the key days for people coming into this country for the Olympic Games. I believe it is not right for them to hold a strike. They will risk damaging people’s enjoyment of coming through into the UK.
Prime Minister David Cameron, visiting Kabul, Afghanistan, also criticized the union.
“My understanding is there was a 20 percent turnout, so I hope it will not take place, he said. I don't believe it would be right or justified.
Josephine Clare Valentine -- Baroness Valentine -- a member of House of Lords and chief executive of London First, a business organization, said: Calling a strike on the demand of 12 percent of members is ridiculous. This action shows a cynical contempt for the public by a minority of PCS members, referring to the union’s low voter turnout.