It is set to move to the disused part of the market in Smithfield in 2022 and will create 1,700 new jobs in the city, according to the museum's creators .
The City of London Corporation will donate £110m to the project, with the Mayor using a loan payable over 25 years to fund the rest.
Planning applications are to be drawn up for 2018 and the museum hopes by the time its complete, the attraction will draw in around two million visitors a year.
Mr Khan said: "From the outset of my mayoralty, I pledged to make culture a core priority and I’m proud that this is the biggest ever cultural investment made by any mayor of London to date."
The museum, currently at London Wall near the Barbican, was established in 1976 and tells the story of the city from 450,000BC.
Museum director, Sharon Ament, said: "Working with our design team we can now move forward confidently with detailed plans for the new museum and remain firmly on target to open the new museum in 2022."
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