Flames tore through Newsome Mills, a four-storey former textile mill in Huddersfield, following a suspected arson attack on Thursday.
The gutted mill has been mostly demolished but a Facebook group wants the tower restored to "former glory".
Councillor Andrew Cooper said: "People have a lot of affection for the mill."
The Green Party councillor on Kirklees Council added: "At the very minimum we need to keep it in the landscape.
"Substantial historic elements are still there and there is a lot of real love and enthusiasm for the building."
Mr Cooper is also a member of the new Facebook group, Save the Clock.
Helen Kingston, who started the social media group, said  "this mill is engrained in all our childhood memories".
The mill site is Grade II-listed and was founded in 1827. It was a working woollen textile mill until 1983.
Most of the buildings were let out as business units until the site was sold in 2006, according to Historic England.
The majority of the building was demolished to the ground floor on Friday.
A landmark clock tower which forms part of a listed mill building gutted by fire should be saved, according to campaigners.