Wandsworth Council could raise council tax by £958 for a Band D property - an increase of 94% - under proposals described as "outrageous".
The Conservative-led London borough said the rise was needed unless the government reconsidered cuts to its funding. Due to changes announced earlier this year, the council said it stood to lose £84m annually in funding by the end of the next three years.
Plans to increase council tax were heard at an extraordinary council meeting on Tuesday but a final decision will not be made until early next year.
The Ministry of Housing, Communities and Local Government (MHCLG) said the changes were needed to reform the "outdated" council funding system.
Wandsworth has a long history of setting low council tax. While an increasing number of London boroughs have seen council tax on a Band D property rise above the £2,000 mark, in Wandsworth, council tax on an average Band D property is currently £1,020.35.
Of that figure £509.84 is the part that the council sets and £510.51 of that is the Greater London Authority (GLA) precept set by the mayor of London.
Wandsworth Council, along with Westminster City Council, Richmond Council and the Royal Borough of Kensington & Chelsea, is currently consulting on plans to use a piece of legislation called the Sustainable Communities Act to try to challenge the government proposals that would see its central government funding reduced.
Changes to the funding formula for councils under the Fair Funding Review 2.0, external will see money redistributed from some council areas to others.
After deciding to reduce their share of central government funding, the government gave five London boroughs - Kensington & Chelsea, Westminster, Wandsworth, Hammersmith & Fulham and the City of London - and Windsor and Maidenhead Council in Berkshire the power to raise council tax above the usual 5% cap for two years.
At Tuesday's meeting, Conservative deputy leader and cabinet member for finance, Peter Graham, said he was reluctantly considering raising council tax and blamed the government.
Graham said: "This administration recognises the financial pressures that many households already experience and the extent to which they are already taxed by other means.
"Higher council tax will not be easy or welcome. We do not regard it as fair and will continue to challenge the government by every possible means.
"Residents can help us with that challenge by responding to the consultation that was launched in the last week or so."
The leader of the opposition, Labour councillor Simon Hogg, said: "Wandsworth Conservatives have decided to double your council tax.
"It's outrageous, it is stunning. People who don't know how to run the council panicking and making you pay the price.
"This increase is unfair, it's unnecessary and we oppose it."
The Labour MP for Putney, Fleur Anderson said: "This huge council tax increase threat will be hard to afford and very worrying for families across Wandsworth.
"The Conservative-run council should listen to residents and reverse this huge council tax increase.
"Wandsworth families should not be forced to pay the price for the cruel choices this Conservative administration has made."
A MHCLG spokesperson said: "We have consulted extensively on reforming the outdated council funding system to ensure money goes where it is most needed, following on from a long-standing consensus... We're also protecting councils through transitional arrangements so no area faces a cliff edge."
A final decision on how much local people will pay in council tax will not be made by the council leadership until February.
