And tomorrow councillors are expected to agree to contribute £500,000 to the cost.
This year, audience figures at the city centre cinema hit an all-time high of 200,000.
Council business manager Alistair Watson said: "GFT is bursting at the seams, regularly taking off films while they are taking considerable levels of box office income.
"It lacks the space to grow the audience for important often issue-based or local community interest films.
"The lack of screens is particularly impacting on its international ambitions for the Glasgow Film Festival."
Ewan McGregor is honorary patron of the GFT's successful seat sponsorship scheme – sponsors have included movie icons such as Steven Spielberg and Orson Welles.
The work will involve converting the existing cafe to a 60-seat cinema, developing a new cafe at the rear of the building and improving disabled access by installing lifts to the first floor.
It is expected to be carried out over 26 weeks, from April to September next year.
The cinema will close for matinees from Monday to Friday but will offer a full service during the evening and at weekends.
The extra cinema will allow for 1500 new programming slots a year, a rise of 50%.
Council leader Gordon Matheson said Glasgow "has to ensure we continue to support and give a platform to new talent" in the film and TV industries.
Jaki McDougall, chief executive of Glasgow Film which runs GFT, said: "A new cinema screen would allow us to increase our Festival audiences."