The two sides met throughout the weekend under the guidance of the conciliation service Acas, and further talks were to take place on Monday.
Flybe, British Airways, Air Canada, Swiss, Lufthansa, Etihad, Qatar Airways, Aer Lingus, and TAP Air Portugal are among those to have confirmed cancellations.
Unite said in a brief statement: "Tomorrow's strike by Unite members working for Heathrow Airport has been suspended to allow for further talks at Acas tomorrow".
Unite said that some 88,000 passengers could be disrupted if 20 per cent of flights are cancelled on Monday and Tuesday, amounting to compensation of 2.3 million pounds (S$3.9 million dollars) for the airlines affected.
A planned two-day strike for Monday (today) at Heathrow Airport has been postponed to allow more time for negotiations with managers.
The union includes workers from security staff, firefighters, engineers and other passenger service operators.
Heathrow said: "We remain hopeful that we can find a resolution and stop this disruptive and unnecessary threat of strike action".
Strike action on Tuesday 6 August is still scheduled to go ahead.
"We regret any inconvenience this may have caused and urge all guests to monitor our website and social media channels for up-to-date information on the status of our flight operations", the national airline company said.
"Airlines may choose to reinstate impacted flights". While 16 were cancelled, five were moved to other London airports.