The Observer reported on Saturday that campaigners want an investigation into what they regard as "misuse of public funds".
So far £60 million of public money has been pledged – half each by Boris Johnson and George Osborne and it was revealed last month that London would underwrite the long-term £3.5m annual maintenance costs if these cannot be raised privately.
This was despite the bridge originally being conceived of a private project, funded entirely by private money.
Michael Ball, director of the Waterloo Community Development Group, who is leading the judicial review claim which looks at planning permission with the help of law firm Leigh Day, has said that the cost of the bridge could fund 30 new London parks.
He has also spoken to Margaret Hodge, chair of the public accounts committee, about a possible investigation over the use of Treasury money for the project after the General Election, the Observer reports.
However the charity which was established to build and maintain the bridge, The Garden Bridge Trust, maintains that the £60m of public funding comes from dedicated transport funds and will be used to provide something which is recognised as having a transport benefit. It is also a small fraction of the £175m needed, £125m of which has already been pledged, according to a spokeswoman. Glencore and Citigroup are among the donors.
She said: "We're extremely confident that we will be able to reach the £175m to be able to build the bridge and the annual maintenance costs."
A statement by The Garden Bridge Trust added: "We note that permission to proceed to a judicial review hearing has been granted on only one of the two grounds sought. We are working closely with the LB Lambeth to discuss the next steps following the decision, so that we can continue to work toward construction start on the Garden Bridge early next year."
Campaigners object to the use of public funds but also a range of other issues, such as location of the project, riverside overcrowding due to popularity, restricted public access and loss of views.
But despite a vociferous public campaign, the bridge retains support from significant bodies. Last month the RHS and The Courtauld Institute of Art reiterated their support for the project, alongside business groups from both sides of the river.
Chairman of the Trustees of the Royal Botanic Gardens, Kew Marcus Agius, who is supporting the project in a personal capacity, said: "The Garden Bridge is a brilliant way to capture London’s 21st Century zeitgeist.
"It will also provide a compelling opportunity to connect residents and visitors of all ages with the crucial importance of plants."
RHS executive vice president Jim Gardiner is a Garden Bridge Trust Trustee.
Chief executive of the Northbank Business Improvement District Ruth Duston said: "The Garden Bridge will be transformational in how people move around this area. The new crossing will be a landmark in London, connecting two of the capital’s historical and cultural hubs – the Northbank and South bank. It’s going to help us put the Northbank on the map and complement the work we are doing to make the area a more attractive place to visit."
Director of The Courtauld Institute of Art Professor Deborah Swallow welcomed the "huge boost for our local businesses" which the bridge would provide by attracting more visitors to the north bank of the river where The Courtauld Gallery and Somerset House are located.
The bridge has been designed by Thomas Heatherwick with engineering by Arup and a planting design by Dan Pearson. It has received planning permission on both sides of the river and approval by Johnson’s office to go ahead.
Frosts Landscape Construction, Willerby Landscapes and Gavin Jones have been shortlisted in a tendering process currently underway. The Garden Bridge Trust expects a tendering decision around mid to end of May for supply, installation and maintenance of plants.