Up to 150 mostly part-time jobs were put at risk last month when loss-making Manston Airport revealed it was in talks over a possible closure.
It was bought by the co-founder of the Stagecoach Group, Ann Gloag, for Â£1 from the New Zealand company Infratil in October.
Since then it is understood the airport has been losing about Â£10,000 a day.
A 45-day consultation period was due to end on 11 May, but this will now not happen, Unite said.
Ian McCoulough, Unite regional officer, said: "We had talks with the management today and it said that the consultation period was now extended until a final decision is made.
"This is to be welcomed."
He said the company was still considering two business plans drawn up by workers at the airport.
Dutch airline KLM said it would not return to the airport even if a buyer is found.
The Dutch airline started running two daily return flights from Manston to Amsterdam's Schipol airport at the beginning of April 2013.
Its last flight was on Wednesday.
KLM Cityhopper managing director, Boet Kreiken, told Air Transport World that it was impossible to do "business in a shaky environment".
"Now it is game over; we will redeploy the aircraft. We are gone."
In November 2011, Flybe announced it would be pulling out of Manston, blaming the size of the airport's catchment area.
The airport is still operating cargo flights.
The consultation period over the future of a Kent airport has been extended, the Unite union has said.