The project, which would transform Festival Square in the city’s west end, has secured their backing weeks after new designs for the nine-storey complex were unveiled.
Swinton, the Highlands-based star of Michael Clayton, Young Adam and We Need To Talk About Kevin, described the proposed new home for the Filmhouse cinema and the city's film festival as an “Enlightenment House” for Edinburgh.
Swinton, who lives in Nairn, said the prospect of the development was a "bright and heartening horizon" at a time when cinema was being "craved" by audiences like never before.Edinburgh-born Glen, whose other screen roles include appearances in Downton Abbey, Prisoners' Wives and Titans, credited the Filmhouse with nurturing a childhood fascination with cinema.
He predicted that its new building would become “a vital creative hub” for the Scottish screen industry if it was given the green light.
The Centre for the Moving Image (CMI), which runs both the Filmhouse and the film festival, unveiled plans for a new home last March after 30 years of discussion and debate over alternative sites around the city.
The building was reduced in height by two storeys by the time final designs were lodged with the city council in December.
Although it have six storeys above ground and is envisaged to be taller than the neighbouring Sheraton Grand Hotel and the Usher Hall, the CMI hopes to secure planning permission later this year, which would allow work to start in 2023 if funding can be raised.
Swinton, a patron of the film festival, said: "At a time when we crave cinema like seldom before - its connections, its inclusiveness and internationalism, its thrills and compassion, its fantasy and transport, its narratives, its dreamscapes, its nourishment, its galvanising perspectives - the prospect of a new Filmhouse for Edinburgh is a bright and heartening horizon.
“Edinburgh, her cine-dedicated populace and her countless perpetual visitors deserve a new Filmhouse, a new lighthouse, a new Enlightenment House to reflect the open eyes of this great capital city and her global outlook.
“This joyous proposal is a great good thing for us all to look forward to, to dream about and to start celebrating now."
Glen said: “As a kid growing up in Edinburgh and becoming fascinated in film, the Filmhouse always had the releases that I most wanted to see.
"I fully support the development of The New Filmhouse as a vibrant creative hub for the film industry in Scotland.”
The new complex has already secured the backing of author and screenwriter Irvine Welsh, whose latest film, Creation Stories, which charts the life of Scottish music mogul Alan McGee, is due to be unveiled within weeks.
Speaking after the latest plans were unveiled last month, he said: “Edinburgh can’t remain bereft of cultural ambition.
"A custom-built Filmhouse would put the city on a par with some of the great cinema capitals of the world.
"It would be a marvellous resource for our community and provide a fitting home for the world’s oldest international film festival.
"Why settle for being an also-ran when you can be one of the very best?”