The sale was announced last year when it was believed that the government would keep a significant stake in the bank, which was launched in 2013 and was the first of its kind in the world.
However, the government is now expected to sell the entire bank, retaining just a token share.
The sale is worth about £2m and Macquarie is believed to have topped a rival bid from a consortium including Lloyds Bank and Sustainable Development Capital.
British entrepreneur Sir Richard Branson is concerned about the government relinquishing control of the bank. At the weekend he said: “The UK Green Investment Bank is so important because it is a pioneering, government-backed national institution that has led the way in green and low carbon finance.
“It is a flag carrier and it is only just getting started. That’s why it’s vital that it remains government backed and green post-privatisation, preferably with government retaining a stake.”
Since its formation the bank has backed 82 low-carbon infrastructure projects, the most recent announced last week and involving £28m of debt finance for an energy-from-waste plant in Scotland.