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.

They were part of the cargo on the SS Politician, which sank off Eriskay in the Outer Hebrides in 1941.
Scotch Whisky Auctions, which sold the bottles, said they had gone to a buyer in the UK after worldwide interest.
They were among eight bottles recovered from the wreck in 1987 by Donald MacPhee, from South Uist.
The whisky is not thought to be fit for human consumption.
The SS Politician was headed for Jamaica with 28,000 cases of whisky when it ran aground on the northern side of Eriskay in bad weather.
Islanders recovered hundreds of cases of whisky from the wreck and some of the bottles were buried to keep them hidden from customs officers.
Other bottles have since been found washed up on the island's shores and also recovered by divers.
Scottish author Compton Mackenzie, published the novel Whisky Galore in 1947, which was loosely based on the SS Politician wreck.
It was adapted for the cinema in a 1949 Ealing comedy starring Basil Radford.

The Toffees have been working with Liverpool City Council to find a replacement for Goodison Park, where they have played since 1892.
Everton chief executive Robert Elstone said the plan was at "an early stage".
He hoped the new sites would provide a "more straightforward, deliverable opportunity" to build a new stadium.
Iranian businessman Farhad Moshiri bought a 49.9% stake in Everton in February and quickly outlined plans for a move from Goodison, which has a capacity of 39,572.
"Goodison Park has served the club extremely well but we need to make sure the club has a suitable stage to perform on for the future," he said.
An initial plan was to build a stadium at Walton Hall Park, a short distance north-east of Goodison, but it was strongly opposed by local protestors.
Mayor Joe Anderson said the site would now remain a park.
"Most people will be aware that I did give a commitment to Everton to support a potential scheme at Walton Hall Park with the aim of regenerating the area and creating new jobs," he said.
"However, through the work that the club and the council have done, we have concluded that effectively building a new village in north Liverpool with lots of retail space is a step too far in this current economic climate."
Across Stanley Park at Anfield, Liverpool are in the process of expanding their main stand which will take their capacity to around 54,000.
A landmark clock tower which forms part of a listed mill building gutted by fire should be saved, according to campaigners.
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Two bottles of whisky salvaged from the shipwreck that inspired the book and film Whisky Galore have been sold for £12,050 after an online auction.
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Everton have abandoned plans for a new stadium at Walton Hall Park but have identified two possible alternative sites within the Liverpool boundary.