Dave Sharman, a Green Party candidate for the Peninsula ward, has written to Greenwich Council asking for assurances that East Greenwich Library will not be closed.
The library, donated to the community by philanthropist Andrew Carnegie, is also home to Greenwich Community College's Meridian Music Centre.
Greenwich Community College is currently undertaking a consultation process on the courses it can afford to offer after it had a 10% cut imposed on its "Adult Learner Responsive" budget this year.
The review involves all aspects of the college, including the Meridian Music Centre, although local Greens claim that staff have already been told the music centre will close at the end of this academic year.
Dave Sharman commented: "I hope the music college's planned closure is not part of a wider project to do away with the library by stealth - the people of East Greenwich will not stand for that"
Cllr John Fahy, Greenwich Cabinet Member for Culture and Olympics, commented that "the Scheme [to refurbish the library] and Estimate is being costed and work will commence shortly".
He added that he would continue to work with the Friends of East Greenwich Library and Ward Councillors "to maximise a long term future for the building".