It said it recognised the educational value of the Sighthill Stone Circle and its potential as a visitor attraction.
However, while the authority wants it to remain, sources say that, 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.
Mr Lunan said: "There's now a recognition of its scientific and historical significance, and its potential as an educational feature and visitor attraction."
A council spokesman said: "We want these stones to stay in Glasgow, ideally in Sighthill, to be used as an educational tool. We have not properly communicated this before and are sorry for that."