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The former police headquarters at Fettes and the site of the former Royal Victoria Hospital in Comely Bank emerged as possible options for the new school after talks between Education Secretary Shirley-Anne Somerville, city education convener Ian Perry and vice-convener Alison Dickie.
Gaelic parents have argued for a stand-alone school near the city centre as preferable to the council’s proposal for the Gaelic school to be built on a joint campus with a new Liberton High School.
Council leaders were taken by surprise when the SNP manifesto at the Scottish Parliament elections in May promised a stand-alone Gaelic secondary for Edinburgh in a central location.
After the talks with Ms Somerville, the council confirmed the Fettes and Royal Victoria sites – which had previously been looked at but dismissed – would be re-assessed.
But the authority made clear that Scottish Government funding of at least £48 million would be needed for the build cost alone, as well as support to secure either site.
Cllr Perry said the talks with Ms Somerville had been “very constructive and helpful”.
"It must be stressed, however, that significant government support would be required to make these viable options. Our number one priority remains identifying the best and most workable solution for a fully immersive Gaelic Medium Education (GME) high school for the city.”
The council is now awaiting a government response.
Edinburgh has had a dedicated Gaelic-medium primary school, Bun-sgoil Taobh na Pàirce in Bonnington, since 2013.
The existing secondary-age GME unit needs to move from its base at James Gillespie's High because the school is at capacity with pupil numbers set to grow.
But a survey last year found the Liberton joint campus proposal was backed by just 15 per cent of Gaelic parents.
Many parents argue Gaelic education thrives better in stand-alone schools where children are in a "linguistic bubble".
And they also say Liberton is not the most easily accessible location from other parts of the city.
Some more central sites were floated as possibilities, including the Princess Alexandra Eye Pavilion on Chalmers Street, the old Royal High School, the Lothian Buses depot on Annandale Street and the council's former depot at Russell Road.
But only the Fettes and Royal Victoria options are now being considered. Both were previously seen as prime sites for new housing.
Cllr Dickie said: "We’re fully committed to ensuring GME can grow and thrive in Scotland’s capital and we have welcomed the Scottish Government’s continued encouragement about our plans for a new and dedicated school at Liberton.
"It is vital though that we get this absolutely right for children and their families, and for the growth of GME in the city.
"Given the growing appetite for a more central location, we have made a decision to pause to re-evaluate previously visited options and their related barriers.
"As well as the substantial financial hurdle involved, we’d need certainty that the land would be available to meet the timescales involved.”