The Russian communications ministry says it will dismantle and relocate the Shukhov tower, a masterpiece of design often compared to the Eiffel Tower in Paris.
But campaigners fear the tower will simply be demolished because it will be impossible to dismantle it and put it back together again.
It is the first time a campaign to save a historic building has received such a wide public following, not just from among architecture enthusiasts.
Government and city officials are set to meet campaigners for talks on the tower this week that may decide its future.
The campaign is not just about saving Moscow's crumbling early Soviet architecture, but also about people having a say in government decisions.
- 'Learn how to say no' -
"We have to learn how to say no," said Moscow resident Anna Chernobylskaya, holding a placard at a recent protest under the tower.
Vladimir Shukhov, a gifted engineer, built the tower to give Soviet Russia a strong radio signal. At 148 metres (485 feet), it was Moscow's tallest structure until the 1960s.
The conical telescope-like design was built with each section lifted into place from the inside. Its slanted supports used the minimal amount of steel.
While often compared to the Eiffel Tower, its design is purely functional.
"It's absolutely ascetic but at the same time very beautiful," enthused Alexandra Selivanova, head of the Centre of the Avant-Garde at Moscow's Jewish Museum, who updates a Facebook page on the tower and has organised several protest events.
The Shukhov tower's owner, the communications ministry, says its poor, corroded condition means it must be dismantled, after which it could be rebuilt elsewhere, with proposals including moving it from its southern Moscow neighbourhood to Sevastopol, Crimea's Black Sea fleet base and territory annexed last month from Ukraine.
"It's the same thing as saying the Eiffel Tower is surrounded by buildings, let's move it to Marseilles," said the engineer's great grandson, who is also named Vladimir Shukhov.
- 'Monument will be destroyed' -
"As all specialists say, the tower can only be taken down one way, sawing it down, and after that the monument will be destroyed," he told AFP.
He said the tower apparently fell victim to the interests of developers who have set their sights on the neighbourhood dotted with several other Constructivist buildings in various states of disrepair.
"This is about redeveloping this whole area," he said. "It's billions (of dollars)."
Architects Norman Foster of Britain and Rem Koolhaas of the Netherlands and Nicolas Serota, the director of Britain's Tate Galleries, were among thousands to sign a petition asking President Vladimir Putin for the tower to be restored in situ.
But Selivanova said that it was not just the architectural community that signed the petition; there were many average people who even posted "selfies" snapped against the landmark's silhouette in an online campaign.
"We've collected almost 5,000 signatures just from ordinary Muscovites," she said.
- Constructivist influence -
The tower is a leading example of the Soviet Constructivist movement of the 1920s and 1930s which focused on function and experimented with new geometric forms, a pioneering aesthetic that eventually inspired generations of architects around the world.
Smaller campaigns are underway to save Constructivist buildings because they are generally in a very poor state of repair after decades of neglect or else are being remodelled in a way that ignores their historic significance.
Many have already been torn down by developers and replaced with new buildings.
There is meanwhile a family versus government row over the private house of architect Konstantin Melnikov, built in the late 1920s of two vertical cylinders dotted with hexagonal windows.
Melnikov's artist son Viktor, on his death in 2006, left the house to be turned into a state museum, and it remains perfectly preserved, even with pencils in original boxes still on Melnikov's desk in the airy studio space.
But the house is caught up in a complex dispute between family members and culture officials over its future, which has prevented its opening to the public.
Viktor Melnikov's daughter and the will's executor, Yekaterina Karinskaya, is living in the house packed with paintings and antiques, and says she does everything from paying heating bills to clearing snow.
"It's a phenomenon of unique architecture, unique spirit," she told AFP during a rare tour. "When you come through the gate, you enter a completely different time."
The culture ministry says it will turn the house into a branch of the Moscow architecture museum, but Karinskaya wants it to become a museum of its own.
While the conflict has dragged on for the past decade, the building is deteriorating.