Glasgow City Council said it recognised the educational value of the Sighthill Stones and its potential as a visitor attraction.
However, while the authority wants it to remain, sources say its future is "in the hands of physics" and if the planned regeneration of Sighthill for Glasgow's 2018 Youth Olympics bid could not work around the structure, it would have to be relocated.
The campaign to save the Sighthill stones – erected in 1979 by astronomer and science writer Duncan Lunan and billed as Britain's first authentically aligned stone circle to have been erected in more than 3000 years – has been gathering momentum since December.
Then, council officials said it was being removed as part of the Youth Olympics bid.
Author and artist Alasdair Gray criticised the plans. Meanwhile, Stuart Braithwaite, guitarist with Scottish rock band Mogwai – whose father helped construct the monument – has attracted support from within the music and arts worlds.
Other backers include Professor John Brown, the Queen's official astronomer to Scotland, and Philip Carr-Gomm, head of the Order of Bards, Ovates and Druids, who has boosted the petition to save the stones to around 3500.
Mr Lunan has met with senior council officials to discuss the future of the structure.
He said: "There's now a recognition of its scientific and historical significance, and of its potential as an educational feature and visitor attraction."
A council spokesman said: "We want these stones to stay in Glasgow, ideally in Sighthill, and we want them to be used as an educational tool.
"We have not properly communicated this before now and we are sorry for that."