New Transport Secretary Chris Grayling is due to decide whether to extend a £15m government underwriting of the £175m project until September 2017.
But tough questions are being asked in Whitehall about the footbridge and its public value, Newsnight understands.
The Garden Bridge Trust said it hoped ministers would continue to support it.
The 366m (1,200ft) tree-covered Garden Bridge, whose construction is yet to begin, would span the Thames between Temple and the South Bank.
Mr Grayling is understood to be keen to study all the options ahead of his decision, which needs to be made imminently to allow the trust - the charity responsible for building the bridge - to file its accounts this week.
The project in central London was championed by Boris Johnson as London mayor and by George Osborne as chancellor.
The concerns in Whitehall about the project, which is due to receive support from the taxpayer to the tune of £60m, focus on two areas amid a feeling that tough questions have not so far been asked:
Sadiq Khan, the new mayor of London, has already said there can be no more support from Transport for London or the Greater London Authority beyond the £30m they are contributing. That is half of the public funds.
A spokesperson for the London mayor told Newsnight: "Sadiq Khan supports the Garden Bridge but he has been absolutely clear that no more TfL or GLA money will be spent on it."
The Garden Bridge Trust has confirmed to Newsnight that it is seeking a meeting with the transport secretary and it hopes the government will continue to support its plan.
But the trust has indicated to Newsnight that the future of the bridge lies in the government's hands because the private sector, which is due to provide more than £100m of the funding, cannot underwrite the project if the transport secretary decides to withdraw support.
The bridge, which was originally conceived by the actress Joanna Lumley as a memorial to Princess Diana. may become a victim of the changing of the guard in Whitehall at London's City Hall.
George Osborne was a great supporter of the bridge on the grounds that it would showcase great British design and attract extra visitors to London.
Mr Johnson became a strong supporter after being lobbied by Ms Lumley who has known him since childhood.
The foreign secretary is now described in Whitehall as a "lone voice" in the cabinet in favour of the bridge.
Mr Johnson's successor as London mayor became a reluctant supporter of the bridge on the grounds that it would be more expensive to cancel the project given that £37m of public money has already been spent.
But Newsnight understands that Sadiq Khan is unlikely to shed any tears if central government withdraws its support for the bridge.
His plans for London have focused on building more bridges to the east of Tower Bridge because they would do more to boost economic development.
The proposed Garden Bridge across the River Thames has run into trouble after changes in the government and the end of Boris Johnson's run as London mayor.