The estimate came as the first £25m stage of repairs got underway this week following detailed investigations by specialist engineers.
The 132-year-old bridge was closed to motorists in April after micro-fractures were discovered in the cast iron casing.
Transport for London and Hammersmith and Fulham Council are now considering funding options for the main repair works phase.
A contract is due to be awarded next spring with the work taking three years.
The council said: “Early stage estimates indicate the work could cost £120m, although as this is purely early estimates it includes a contingency due to the unknowns, complexities and challenges inherent in repairing such an aged, Historic England listed structure.
“Engineers will continue to refine this estimate as the project progresses.”
Cllr Stephen Cowan, Leader of the London Borough of Hammersmith & Fulham, said: “This comprehensive structural review was the first in decades. It has revealed corrosion and significant failings throughout the 132-year-old suspension structure that, had they been allowed to continue unchecked, would have been a threat to public safety.”