The consortium also comprises a UK-based pension fund, the Universities Superannuation Scheme (USS); and a fund managed by Mira, a subsidiary of Macquarie's asset management arm.
The long-time rumoured deal will see Macquarie acquire GIB, which will then manage or supervise £4 billion (€4.78 billion) of green infrastructure assets, including offshore wind projects.
One of the major criticisms the UK government and Macquarie have faced over the sale would be the potential loss of the bank's identity as a backer solely for green or low-carbon projects
Macquarie has pledged to invest a further £3 billion (€3.59 billion) in green infrastructure projects over the next three years and to "maintain its green purpose". The bank will become Macquarie's main investment vehicle in the UK and Europe, meaning it could invest in overseas projects.
A special share arrangement has been agreed, designed to safeguard these purposes, Macquarie said. Five trustees have been appointed to hold this share under a 'Green Purposes Company' (GPC).
GPC, which will control a special share in the GIB once it is privatised, is designed to prevent large changes that could damage the bank's green remit, though the trustees will not have any other voting rights.
Most of the nominees are well known to the environment sector.
The Australian firm said upon completion of the sale, due by mid-2017, GIB will create three investment vehicles: "an offshore wind investment vehicle, a low carbon lending platform and a green infrastructure investment platform".
"We understand the responsibilities that come with this ownership, and we are fully committed to maintaining its green purpose as we grow the business," said Macquarie head of capital in Europe, Daniel Wong.
UK climate change and industry minister Nick Hurd said: "This deal gives us the best of both worlds. We have secured fair value for the UK taxpayer. GIB has a well-funded new owner that is committed to the bank's green mission, with a track record of success in green investment and an ambition to grow the business."
In a statement to parliament, Hurd added: "The government will continue to hold a £130 million portfolio of a small number of GIB's existing investments. This portfolio will continue to be managed by GIB until these investments can be sold on in a way which returns best value for taxpayers' money."
The GIB has supported several offshore wind projects in the UK. E.on's 400MW Rampion, the 630MW London Array and Innogy's 576MW Gwynt y Mor are among the projects to receive support from the bank.
"Macquarie has made significant and important commitments to the UK Government to maintain GIB as a discrete entity within its business, maintaining GIB's investment focus and approach with a target to invest more capital each year than GIB has historically," said GIB independent chairman Lord Smith of Kelvin.
"Macquarie will also uphold GIB's green investment principles and report transparently on GIB's green impact. Macquarie will utilise the market-leading expertise of the existing GIB team and will build on GIB's deep commitment to Edinburgh," Lord Smith added.
The bank was launched in November 2012 by the government, with more than £3 billion of funding.
The UK government announced its plans to privatise the bank in June 2015. The then-chancellor, George Osborne, said the sale would "enable it to access larger pools of capital".
In December 2015 a group of MPs said the government had failed to make a reasonable case for selling off the bank.
In a report, the members of the environmental audit committee said: "The government's case could have been strengthened if it had provided evidence that this is the right time to privatise the Green Investment Bank.
"The absence of this, coupled with the government's reliance on its view that this is merely the 'natural next step' for the bank, is likely to lead to the suspicion that the move and its timing are not evidence-based policy," the report added.