Boris Johnson has claimed Transport for London (TfL) was "effectively bankrupted" even before Covid-19.
The comments come after Sadiq Khan accused the government of "draconian" demands in return for a second bailout.
But the prime minster said any need to increase fares was "entirely the responsibility" of the London mayor.
Mr Khan responded by calling the PM a "liar" and said he had cut the operations deficit, left by Mr Johnson when he was mayor in 2016, by 71%.
A spokesperson for 10 Downing Street told the BBC Mr Johnson "stands by his comments in the House of Commons".
The mayor told the BBC: "It gives me no pleasure in saying so, but our prime minister today, on the floor of the House of Commons, has lied.
"We as a transport authority rely hugely on transport fares from Londoners, and more than 90% of that has dried up, which is why we need a Covid bailout from the government.
"What the government is doing instead is punishing Londoners by imposing all sorts of draconian conditions."
The mayor previously said ministers aimed to impose a "triple whammy" of measures in return for rescue funding to cover the losses incurred through Covid-19.
The proposals include higher council taxes, an expanded congestion charge zone and a hike in Tube and bus fares.
Mr Johnson told the Commons on Wednesday: "The current mayor of London had effectively bankrupted TfL before coronavirus had even hit and left a massive black hole in its finances.
"Any need to make up that deficit is entirely down to him, it is entirely his responsibility.
"Any expansion of the congestion charge or any other measure taken to improve the finances of TfL are entirely the responsibility of the bankrupt current Labour mayor of London."
The 10 Downing Street spokesperson said TfL's debt has increased 30% since Mr Khan became mayor.
"TfL debt is now a record high of £12bn and now spends £402m a year on debt interest," the spokesperson said.
The Department for Transport said talks over a settlement were continuing.
Labour's deputy leader Angela Rayner said: "We now have a prime minister so determined to punish a Labour mayor that he wants to whack a transport tax on his own constituents.
"Yet still they refuse to take the decisive national action needed; instead, they have tried to play people off against each other. Divide and misrule."
Mr Khan told a TfL board meeting: "Now is not the time for the government to play party political games or be vindictive towards London. This is far too serious a matter.
"I intend to stand firm and fight for a fair deal for Londoners and do what's right for our city."
TfL bosses have asked for a £5.7bn package to prop up services for the next 18 months, after passenger numbers and revenues plummeted in the aftermath the March lockdown.
In May, the government granted TfL £1.6bn in emergency funding to keep services running. Then on Friday, a last-minute extension of that bailout was secured to cover another two weeks.
As part of the package of new measures, according to Mr Khan's office, ministers want to:
- Extend the £15 Congestion Charge Zone to the north and south circular roads in 12 months' time
- Impose an above-inflation fares increase
- Introduce a new council tax precept charge in London for an unspecified amount
Union leaders have criticised the proposals, with some highlighting what they say is a discrepancy in the government's treatment of private rail companies and TfL.
By BBC London political editor Tim Donovan
The virus has ripped the heart out of TfL's finances - with fares down 90%.
The government is adopting a tough negotiating stance with the mayor on how money can be recouped and transport should be funded now in the absence of usual income.
There's a condition that Sadiq Khan extend the congestion charge zone to within the north and south circular.
He's being pressured to agree to big fare hikes and removal of concessionary travel for children and pensioners.
A plan has now emerged for a precept - an extra component to the council tax as exists currently for the police.
It would mean the cost of public transport spread across all Londoners whether they use it or not - but apparently no charge for the millions of users who live outside the capital.
There's been a ministerial threat to take direct control of TfL, and there are ominous rumblings from the unions.
A pugnacious and provocative approach is emerging towards Labour-run devolved administrations.
For Boris Johnson, metro mayors are currently proving the real enemy.
Rail, Maritime and Transport union general secretary Mick Cash said: "It is appalling that the government are targeting staff pensions amongst a range of other savage measures in this short-term funding deal.
"I put them on notice that any attempt to hack away at our members' pension rights will be met by an all-out campaign of political and industrial resistance."
A Department for Transport spokeswoman said: "We have agreed an extension to the support period and to roll over unspent funding from the Transport for London Extraordinary Funding Agreement, allowing further time for negotiations for a new settlement.
"These discussions will ensure London has a safe, reliable network. It would be inappropriate to disclose further details at this stage."