The east London site has been blighted by years of industrial use and is the largest patch of derelict and contaminated land in the capital.
The duo will share the workload across the 270 ha site in the lower Lea Valley.
It is understood that Nuttall will be responsible for works across the south side of the park, with Morrison taking on the north side.
The pair beat off rival bids from Kier, Birse, May Gurney and Mowlem to claim the contract.
A source at one of the successful bidders said works could begin on the park as soon as July.
The insider said: 'We will be managing a variety of different projects on the site using specialist contractors for elements such as asbestos removal, soil washing and bioremediation. There is a lot of work to do.'
The contract was advertised by the London Development Agency in April 2005 but responsibility for the deal will transfer to the Olympic Delivery Authority.
The ODA is due to appoint its delivery partner to oversee the construction and legacy Olympic projects next month.
Nuttall and Morrison both declined to comment. An ODA spokeswoman said: 'There will be an official announcement shortly.'
The main alteration to the park involves moving the broadcast and media centre from the Stratford City development land to a larger site in the north-west corner of the park at Hackney Wick, allowing workspace for Hackney Borough Council following the Games.
Mr Higgins said: 'Having a detailed masterplan for the Games at this stage helps us to really bear down on costs and maximise legacy benefits. We do not want white elephant projects.'