The shopping centre is due to be knocked down as part of a £850m redevelopment of the site at the east end of Princes Street.
The revamped site will eventually feature a 210-room hotel, 30 restaurants, luxury apartments and a multi-screen cinema.
The St James Centre site and the adjacent New St Andrew House office complex were built in 1973.
They are located within the Edinburgh World Heritage Site.
The redevelopment is due to be completed by 2020.
In a report to go before councillors next week, it said the majority of compulsory purchase orders (CPOs) have been agreed and negotiations are progressing well for outstanding land acquisitions.
A footbridge which crosses Leith Street is being removed as part of the project but the council, which owns the bridge, plans to store it and try to "recycle" it in a new location.
The development, which will keep the existing John Lewis store, will also include a cinema, which is expected to offer more art house films than the Vue multiplex at the nearby Omni centre.
Edinburgh St James is a 1.7 million sq ft city centre development and one of the biggest regeneration projects currently underway in the UK.
It will provide 850,000 sq ft of retail space, five star hotel accommodation and will deliver up to 250 new private residential apartments.
The new commercial, retail and residential district is forecast to support more than 3,000 permanent jobs.
It will link Princes Street to Multrees Walk in a series of crescents with a retail and hospitality training academy, as well as flats, restaurants and shops.
Andrew Burns, Edinburgh city council leader, said: "The Edinburgh St James development will now be able to take a step forward with the council, Scottish government and TH Real Estate, the developer, finalising the details of this innovative funding package.
"I am delighted the council has played its part in helping unlock £850m of private sector investment at the heart of our city centre.
"Not only will it provide much needed premium retail space for the city, it will also contain 250 new homes, 30 restaurants, a multi-screen cinema and a 5 star, world class hotel.
"We anticipate demolition will begin this May with an expected completion date of 2020."
Demolition of the St James Centre in Edinburgh will begin in May.