Lambeth Council says it will resume negotiations with the Garden Bridge Trust over the use of land on the South Bank after Transport for London slashed its funding for the bridge from £30 million to £10 million.

The council halted talks in September, citing concerns about the £60 million of public funds to be spent on the bridge, half from Transport for London and half from the Department for Transport.

"I'm pleased that Londoners are getting a better financial deal particularly at a time of austerity when all public sector organisations are being forced to make deep cuts to services," said Cllr Lib Peck, leader of Lambeth Council.

"We've been in tough negotiations with the Garden Bridge Trust and Transport for London and I'm pleased we've successfully agreed a deal that will cut London taxpayers' contribution towards the Garden Bridge by two thirds."

Lord Davies, chairman of the Garden Bridge Trust, said: "We are delighted to move forward with the project.

"We have been hugely successful in our efforts to raise funds from the private sector, with £85 million pledged to date, and we have agreed that any of the committed funds from TfL spent over the £10 million will be treated as a loan.

"We are delighted the Garden Bridge can now progress and are grateful for all the support we've had."

Labour's London mayoral candidate Sadiq Khan is claiming credit for the new deal.

"This is a much better deal for Londoners," he said.

"We've been able to secure an agreement which will allow the Garden Bridge to proceed while saving up to £20 million of Londoners' hard-earned money.

"At a time when public services are straining under pressure from cuts and increased demand, Boris Johnson was wrong to commit to £30 million of London taxpayers money to this project.

"But thanks to with this new deal, it looks like Londoners will get the best of both worlds – Lambeth Council and Lib Peck deserve great credit for their part in negotiating it.

"As this project progresses, I'll be keeping a close eye on it to ensure that London's tax and fare payers aren't burdened and that all funds over £10 million are returned as agreed.

"We need to keep the Garden Bridge on track without costs rising – when I'm Mayor, there will be no question of a penny more than £10 million of TfL funds being spent on it."

Liberal Democrat London Assembly member Caroline Pidgeon said: "While I welcome a reduction in Transport for London's payment towards the Garden Bridge, today's announcement simply raises more questions than it answers, especially in light of the continued evasiveness by the Mayor of London and TfL in answering mayoral questions, or responding to freedom of information requests."

She added: "In addition to these basic questions there are also key issues as to whether the proposed Garden Bridge is in the right location. Is the loss of land on the South Bank and the loss of views of St Paul's Cathedral really justified.

"Most importantly the Mayor and TfL still need to come clean as to how key contracts have already been awarded. Only last week TfL's director of audit finally conceded that the Garden Bridge procurement was 'neither open nor objective'. These are serious issues which are far from being resolved.

"Today's announcement should not for one moment cloud the very dubious practices that have been adopted so far in awarding contracts and allocating significant amounts of public money."