The club want to knock down the Edwardian B-listed London Road Primary School as part of their plan to regenerate the area in time for the Commonwealth Games.
The proposal also includes the formation of a landscaped avenue, essentially a wide walkway leading from a revamped Dalmarnock railway station, past several key 2014 venues, to Celtic Park.
Parkhead is hosting the 2014 Games opening ceremony, which will be attended by the Queen.
The club submitted a masterplan, dubbed The Celtic Triangle, almost three years ago with a view to enhancing the area around the ground.
However, although much of the publicly funded work is either done or on the verge of completion, little has been done in the immediate vicinity of the stadium.
A 2009 council report raised concerns the area around the ground "does not provide an attractive setting", adding "the current situation is not acceptable with regard to the council's ambitions to host a world-class event for the 2014 Games".
London Road is understood to be Glasgow's sole remaining B-listed school and the demolition plans is sure to raise objections.
Celtic have still not taken up the option to buy the school from the council's property arm, despite arranging to do so for around £300,000 several years ago. It is understood the club will take up the option if the application is passed.
However, even if demolition is approved, tight timescales and budgets mean the new landscaping and any new structures would not be up and running in time for the Games.
The 2014 organisers will spend £14 million on the opening and closing ceremonies of the Commonwealth Games, with an estimated one billion TV viewers expected to watch the opening at Celtic Park.