Four extra classrooms and more toilets could be built at Carntyne Primary School on Redford Street.
Glasgow City Council is behind the proposals and has said the extension is required to meet increasing demand for school places.
The application, for a two-storey extension to the existing building, states work would take place while the school is operational, with pupils and staff remaining on site throughout construction.
It added: “The original development of Carntyne Primary School offers a community facility that is well integrated within the urban fabric of this residential area.
“As originally planned for, an extension on the western edge of the school, with an addition of four classrooms, is deemed necessary going forward.”
The current building replaced old facilities on the site in 2017 and an extension area was identified in the original planning application.
Brickwork on the extension will match the existing building and the brick type and colour was “originally chosen to reflect the surrounding traditional architecture”.
An online meeting was held in June last year to present the proposal to school staff and the parent council.
Councillors backed extending the school at a meeting in February last year, with around £8m set aside for work on Carntyne Primary and St Paul’s High School, Pollok.
Both projects involve adding more classrooms to meet the demand for school places.
Speaking at the time, the city’s education convener, Cllr Chris Cunningham, said: “Demand for school places across the city has never been so high so we need to make sure that our schools have places for local children and young people.
“These are very exciting plans that will enhance the school estate and benefit children and young people who will flourish in the new builds or school extensions.
“The plans will create learning environments that will nurture learning and teaching for our future generations and meet increasing demands for schools in certain areas of the city.”