The package was found in the security area where passengers are searched before they board a plane, said the spokesman, who spoke on anonymity in line with company policy.
He said the departures area was evacuated as a precaution and bomb disposal squads were called to the scene.
The airport remained open late Wednesday, but three domestic flights were delayed until the explosives experts could declare the departures area safe, Glasgow police said.
There were no reports of injuries and police declined to provide information on whether anyone was arrested.
Security staff at U.K. airports are on high alert for suspicious packages after two mail bombs bound for Chicago were found in England and Dubai last week.
Authorities said the bombs were designed by the top explosives expert working for al-Qaida in the Arabian Peninsula, the Yemen-based militant faction thought to be behind the plot.
Glasgow Airport was the target of a failed terror attack in June 2007, when two men attempted to crash a blazing Jeep loaded with explosives into a departure terminal. The car's path was blocked and the explosives failed to detonate.