Lord Davies, the Labour peer who chairs the Garden Bridge Trust, told the BBC's Newsnight programme that the proposed completion date for the footbridge between the South Bank and Victoria Embankment has now slipped from 2018 to 2019.
The charity is still £55.9 million short of its the fundraising total, but Lord Davies insisted "the pipeline is very strong" and trustees were confident the money would be found.
He conceded that "one or two" potential donors had dropped out.
For the first time the trust has admitted that costs have risen by 6 per cent to £185 million.
The project has already spent £36 million of public money before any physical works have begun. The bridge has received £30 million from Whitehall via the Department for Transport and a further £30 million in grants and loans from Transport for London at the behest of former Mayor Boris Johnson.
Speaking on Newsnight, Lord Davies insisted the bridge would be worth the effort: "It does sum up what is great about Britain ... it will be magical."
Last month Lord Davies wrote to the new transport secretary Chris Grayling urging him to maintain the Government's backing for the scheme.
He offered to visit the Department for Transport with the project's founder Joanna Lumley and designer Thomas Heatherwick to brief the minister on the scheme.
Lord Davies said on Thursday: "This project is majority privately funded so this unique London landmark will be delivered with great value to the taxpayer.
"The Department for Transport (DfT) and Transport for London (TfL) have been valuable supporters and partners from the very beginning.
"They provided funding because of the clear public transport case for the bridge, without which we would not have got the project off the ground. This funding has helped kick start the public funding drive.
"Now is a crucial time for the Garden Bridge. We have faced considerable challenges but we are now on the brink of building a truly unique crossing. It would be a tragedy if the government withdrew their support now.
"The decision now rests with the DfT to extend the underwriting. We are not asking for more public money but we do need the Government's renewed backing."
This week the DfT wrote to several objectors to the bridge who had urged Mr Grayling to halt the project.
Civil servants said that there were no plans to provide extra Government funding for the bridge.
Earlier this week the Mayor of London Sadiq Khan conceded it was possible that the bridge would never be built.
The bridge will be the subject of an adjournment debate in the House of Commons on Wednesday 7 September.
Before construction can start, the Garden Bridge Trust needs to reach agreement with Coin Street Community Builders over the use of land on the South Bank.
The trust also needs to satisfy the Mayor of London that adequate arrangements are in place to fund future maintenance of the structure.
For the latest local news and events direct to your inbox every Monday, you need our weekly email newsletter SE1 Direct.
7,000+ locals read it every week. Can you afford to miss out?
Read the latest issue before signing up