Mr Swinney confirmed the funding ahead of the official opening of the 116th Royal National Mod festival, which celebrates Gaelic culture.
The event has returned to Glasgow for the first time in three decades, and will see more than 3,000 participants take part in a range of competitions.
The new school, likely to be in the north-west of the city, is being built to meet demand for Gaelic medium education.
Mr Swinney, who will address the Mod’s opening event, is expected to say: “Gaelic medium education is central to our efforts to support the language and we must ensure it also has strong links to Gaelic in the home, the community, or the workplace.
“We want to ensure those who wish to learn and use the Gaelic language are given every opportunity to do so, and I believe we are at a crucial turning point for promoting Gaelic education in Scotland.
“In the 20 years since the first Gaelic primary opened, Gaelic education has been highly valued by children, parents and carers. The new Gaelic primary school in Glasgow would undoubtedly allow more children to access the benefits of such an education.”
Scottish Gaelic is considered at risk of dying out and is classed on Unesco’s list of imperilled languages as “definitely endangered”.
The 2011 census showed only 1.7 per cent of people in Scotland had some Scottish Gaelic skills - around 87,100. Of these, only 32,400 were able to understand, speak, read and write it.
Councillor Chris Cunningham, education convener on Glasgow City Council, said: “Gaelic Medium Education (GME) in the city has never been so popular and this demand is a testament to the commitment of the council to preserving and promoting the Gaelic language and culture.
“More and more parents – with and without the language – are choosing GME for their children and young people and the benefits that this brings to a child’s learning.
“We are proud to now be in a position – with the support from the Scottish Government – to engage with families across the city about a fourth primary school in the city to meet the growing demand for GME.
“We’ve been engaging with the parent bodies over the last couple of years and a formal consultation will be launched for all stakeholders to make their views known by the end of this year.”