A ballot over pay involving 6,000 staff belonging to the Unite union resulted in a three to one vote in favour of strike action, the union said.
Unite will meet with its key representatives on Monday to decide what form of industrial action its members will take.
Any action will affect not only Heathrow but also Stansted, Southampton, Glasgow, Aberdeen and Edinburgh airports.
Before the ballot result was announced, Prime Minister David Cameron warned that a strike would achieve nothing "apart from damage".
But Unite national officer Brendan Gold said: "This ballot reflects what our members feel about BAA's current attitude."
The union had urged firefighters, engineers and support and security staff at BAA's six UK airports to vote for industrial action over what it called the Spanish-owned company's "measly" pay offer.
The union said staff had already accepted a pay freeze in 2009 and that this year the company had offered staff a 1% rise, plus 0.5% which was conditional on changes to a sickness agreement.
Unite also wants workers to receive a performance-related bonus which it said was promised to them if the company hit a certain financial target.
BAA said it made a "reasonable" offer at a time when "BAA and its airline customers are seeing a decline in passengers due to the impacts of recession and volcanic ash".
Mr Cameron, who made a speech today about the importance of tourism to the UK, said: "These sorts of strikes never achieve anything apart from damage - damage to business, damage to jobs, damage to the interests of tourists who want to come to visit Britain, or people who want to leave Britain and have a holiday overseas.
"I very much hope that they don't go ahead. They will do nothing but harm. We want to demonstrate that Britain is open for business."
If the strike does go ahead it will mean more misery this year for air travellers who have already had to cope with a long-running dispute by British Airways workers and widespread disruption caused by the volcanic ash clouds.
The Prospect union, which represents around 100 BAA workers based at Heathrow, Edinburgh, Glasgow, Southampton and Stansted airports, said 53% of its members had also voted in favour of strike action.
Brian Boyd, Unite national officer for civil aviation, said: "Last year BAA's employees accepted a pay freeze to help the company because they understood the difficult financial operating environment within civil aviation.
"A pay offer of 1% plus the withdrawal of two payments worth over £1,000 is simply confrontational.
"BAA chief executive Colin Matthews recently hailed BAA's financial performance, and passenger numbers at Heathrow reached a record high in July. BAA has constantly ignored the contribution its employees make to the ongoing success of the business. Unite members have delivered a strong message that they deserve more."
Mr Gold said BAA was "doing passengers a great disservice by allowing this dispute to get to this stage".
He called on BAA "to return to the negotiating table with a fair offer".
Unite balloted 6,185 staff at the six airports. The total number of valid papers counted was 3,054, with 74.1% voting yes to strike action and 25.9% voting no.
A BAA spokesman said: "We regret the uncertainty this vote has already caused our passengers and airline customers.
"We hope that the union will engage with us quickly to conclude an agreement.
"Fewer than half of those people eligible to vote have done so and we do not believe this result provides a clear mandate for strike action."
Announcing the ballot result at a media conference in London, Mr Boyd said Unite had a clear mandate for strike action and that BAA "faces a total shutdown of its six UK airports".
He and Mr Gold stressed that no decision on strike action would be taken before Monday, there was still time for BAA to "get round the table and negotiate" and that it would "not take much" to sort out the dispute.
Asked if the union was trying to ruin people's holidays, Mr Gold said: "That's not our intention."
Asked if he had a message to the travelling public, Mr Gold replied that he hoped they would want to ask "why BAA has let this run on for four months".
Mr Boyd said strike action was a last resort, while Mr Gold said the aim was still to get a "negotiated settlement".
Both men denied any suggestion that the low turnout in the ballot weakened the union's position.
"The vote was substantially in favour of strike action among those who did vote," said Mr Boyd.
A spokesman for Manchester Airport said: "In what is likely to be a busy period for holidaymakers, Manchester Airport will remain open and our normal schedule will be unaffected. We are already in talks with airline partners to take additional capacity if they require it during the strike period that will affect BAA airports."
The Public and Commercial Services (PCS) union will announce tomorrow whether its members - which include workers who manage terminals and security staff - have voted in favour of action short of a strike. Up to 400 members were being balloted on whether they wanted to work to rule, putting a temporary halt to any overtime. The result is expected to be announced at 1pm.
Transport Secretary Philip Hammond said: "Unite need to consider very carefully the result of this ballot - with only about 35% of staff that were balloted voting in favour of strike action.
"As the Prime Minister has said, these sort of strikes never achieve anything apart from damage - damage to business, damage to jobs, damage to the interests of tourists who want to come to visit Britain and people who want to have a holiday overseas.
"At a time when we are focused on getting Britain back on its feet I would urge both sides to seek an amicable solution."
Comments