Transport for London (TfL) had wanted to provide £30m for the proposed £175m bridge set to stretch from Temple to the South Bank in Lambeth.
But talks broke down after Lambeth Council said not enough had been done to attract private sector backing.
Chair of the Garden Bridge Trust, Lord Mervyn Davies, said he was "delighted to move forward with the project".
In September, Lambeth Council halted negotiations and said it would only proceed if assurances were made that £30m was not taken from Transport for London.
Lambeth council leader Lib Peck said: "I'm pleased we've successfully agreed a deal that will cut London taxpayers' contribution towards the Garden Bridge by two thirds."
Plans for the bridge were approved in 2014, after Lambeth and Westminster City Councils granted planning permission with the chancellor and London Mayor Boris Johnson pledging £30m each, and the rest due to come from private funding.
Lord Davies said: "We have been hugely successful in our efforts to raise funds from the private sector, with £85m 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."
But Liberal Democrat Caroline Pidgeon from the London Assembly 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.
"Why did TfL ever agree in the first place to contribute £30m to the Garden Bridge?  How much will TfL be willing to lend the Garden Bridge Trust?"
The Trust said it hoped to begin working on the site in the New Year.

Pte Manning, formerly known as Bradley, was sentenced to 35 years in military prison for leaking a massive trove of classified US documents.
After the conviction, she announced the desire to live as a woman.
However, the US military prohibits transgender people from serving openly in the military.
Rear Admiral John Kirby, the Pentagon spokesman, confirmed to the New York Times that Defence Secretary Chuck Hagel had approved a request from the Army to "evaluate potential treatment options for inmates diagnosed with gender dysphoria".
The Associated Press news agency first reported the US military was weighing a potential transfer to civilian prison, citing unnamed Pentagon sources.
But on Wednesday, Rear Adm Kirby said no such decision had been been made yet.
"Any such decision will, of course, properly balance the soldier's medical needs with our obligation to ensure Pte Manning remains behind bars," he said.
A local judge in Leavenworth, Kansas, approved Pte Manning's name change request last month, a move the military did not oppose.
Pte Manning has been diagnosed by military doctors multiple times with gender dysphoria, the sense of one's gender being at odds with the sex assigned at birth.
But she has requested treatment, including hormone therapy, and the ability to live as a woman.
According to a complaint filed by Pte Manning's lawyer, a military doctor at Fort Leavenworth  - where she is being held - had approved a treatment plan by November 2013.
But it was delayed as it was sent higher up the chain of command for consideration.
The US military is required to treat diagnosed disorders of its soldiers but its policy allows summary dismissal of transgender people.
Mr Hagel has said the military policy on transgender soldiers "continually should be reviewed".
"Every qualified American who wants to serve our country should have an opportunity if they fit the qualifications and can do it," Mr Hagel said on Monday, but he did not say if he believed it should be overturned.
Despite this policy, a recent study by a US university estimated there were about 15,000 transgender people serving in the US armed forces.
Pte Manning will not be discharged from the military until she finishes her sentence. A judge recently denied a clemency request.
Transfers from military prisons to civilian Federal Bureau of Prisons facilities are not unprecedented, but they are usually done after the inmate has been discharged from the military.
Financial contributions to the Garden Bridge by London's public transport network have been limited to £10m.
[NEXT_CONCEPT]
The Pentagon is considering transferring Private Chelsea Manning to a civilian prison in order to treat her gender dysphoria, US media report.