The contractor bought the site back in 2008 for around £14m.
Balfour has invested almost £30m in the site since then, but passenger numbers have dropped over the years and the airport is reported to be running at a £1.5m loss.
It decided to put the airport up for sale a month ago as part of a plan to raise cash from offloading assets.
In a statement Balfour said: “We regretfully confirm that if no agreement can be reached before 7 October 2014, which ensures the viability of its operations, then it is likely that the airport operations will close.
“In this event, we expect that the last commercial flights will take place on Wednesday, 15 October 2014.”
Cllr John Jones, Blackpool Council’s cabinet member for highways and transport, said: “We have been aware that Balfour Beatty have been making significant losses at the airport, however, the speed of the decision comes as a shock and we are naturally disappointed.
“Our first concern is, of course, for customers and those who are overseas or are due to be overseas at the time of the intended closure date, as well as those who have existing bookings.
“We will be seeking answers from Balfour Beatty and the airlines over these issues immediately.”
He added: “We will now work with Balfour Beatty as well as Fylde and Wyre Councils, to maximise the value of the site to the Fylde coast and look at how it can contribute to creating jobs and supplying and complementing industry.”