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."

Panathinaikos led 1-0 in a play-off to secure a place in next season's Champions League.
On 55 minutes, with the sides each down to 10 men, a melee saw two more sent off.
Moments later, PAOK coach Vladimir Ivic fell to the ground after being hit by a bottle thrown by a fan and the game was halted before being abandoned.
The 40-year-old was struck on the right side of the head and received treatment in the dressing room at Stadio Apostolos Nikolaidis in Athens before having stitches in hospital.
After close to an hour's suspension, referee Giorgos Kominos abandoned the match.
"The natural and moral perpetrators... will be hunted with all the means of redress," PAOK owner Ivan Savvidis said in a statement.
PAOK, Panathinaikos, AEK Athens and Panionios, are in a round-robin play-off to determine who joins league winners Olympiakos in the Champions League.
Panathinaikos' Odisseas Vlachodimos saw red after 20 minutes with his side a goal up but Amr Ward picked up two yellow cards in four second-half minutes to even up the numbers.
Three minutes later, ex-Barnsley goalkeeper Luke Steele was kicked by PAOK's Jose Angel Crespo, who then grabbed Steele's team-mate Marcus Berg by the throat and threw him to the floor.
This led to pushing and shoving by several players, prompting red cards for Panathinaikos' Mali defender Ousmane Coulibaly and PAOK's Ukraine midfielder Yevhen Shakhov.
Demolition of the St James Centre in Edinburgh will begin in May.
[NEXT_CONCEPT]
Four red cards, a mass brawl and a manager struck on the head by a bottle - little wonder then that Panathinaikos v PAOK Salonika was abandoned.