Talks also involve the possible purchase of sister airports at Aberdeen and Southampton with all three currently owned by London-based Heathrow Airport Holdings.
A mystery buyer has been in discussions with HAH officials amid speculation it could be HAH's biggest shareholder - the Spanish transport infrastructure giant Ferrovial.
Talks are thought to have been going on for months after a potential rival walked away from the negotiating table. A consortium led by the Switzerland-based Partners Group withdrew a few months ago.
The Swiss were backed by council leaders in Glasgow with financial support promised by the Strathclyde Pension Fund - the richest council pension fund in the UK with assets totalling £13.5billion.
Ferrovial owns a 25% stake in HAH and had been expected to splash out £800million to secure total control of the business and ownership of the three airports. It already owns Heathrow.
Airport workers were told by e-mail the airports had been put up for sale and negotiations had already started with an unnamed party.
One worker said: "We are under strict instructions not to talk to you people in the media and that their own press team are dealing with inquiries. But obviously the uncertainty is worrying for us all."
Glasgow Airport employs around 450 workers and supports more than 7000 others across Scotland.
There had been an angry reaction from staff earlier in the year when the Evening Times revealed initial attempts of a buy-out by the Swiss-led consortium.
Last night a statement from Heathrow officials, said: "Over recent months Heathrow Airport Holdings group shareholders and management have been considering their strategic position in relation to our three airports, Aberdeen, Glasgow and Southampton.
"As a result the group is now formally entering a sale process. While there is currently no certainty that a sale will be concluded, the group intends to work towards completing a transaction by the end of the year."
A spokeswoman for HAH said: "We can't really comment beyond our statement because it is very commercially sensitive at the moment."