If the walk-outs go ahead, Unite says, almost 2,500 staff will miss work.
Heathrow is yet to announce which flights will be cancelled, and said passengers should check with their airlines to see if they were affected.
Affected passengers might be offered other flights or refunds by their airline, it added.
Heathrow said the flight cancellations, which affect 91 airlines including British Airways, were a pre-emptive measure in case a solution was not found.
Talks between union leaders and Heathrow management at the conciliation service Acas, aimed at averting the strike action, lasted until late on Friday and resumed on Saturday.
The airport, which advised passengers to check its website for updates, said its contingency plans would keep Heathrow open and safe on both strike days, albeit with some disruption.
Passengers still scheduled to fly on Monday and Tuesday have been warned to arrive at least three hours ahead of long-haul departures and two hours ahead of short-haul departures, because it may take longer to get through security.
A Heathrow spokeswoman said: “I can confirm that we are working with our airline partners to consolidate and reduce the number of flights operating during the strike period.
“We have proactively cancelled 177 flights departing Heathrow across Monday and Tuesday.
“Passengers on these flights will have either been rebooked onto alternative services or provided a refund.”
services or provided a refund.”
Unite regional co-ordinating officer Wayne King said: “This latest vote for strike action points to growing anger among the airport’s workers in a whole range of vital jobs which are essential to the smooth and safe running of Heathrow.
“Airport bosses need to heed this latest strike vote and the overwhelming rejection by our members of the revised pay offer which offers little over and above the original offer of £3.75 extra a day for many workers.”
Meanwhile, talks aimed at averting a separate strike by British Airways pilots are to continue next week.
Leaders of the British Airline Pilots Association (Balpa) met the company last week to try to resolve the dispute over pay.
The union would have to give two weeks’ notice of any industrial action