The project will see it create a 3,000m² extension to an existing tower block, which will be set within a large public space.
It will provide a new centre for creative and digital industries and has been made possible by a £5m grant awarded by the local enterprise partnership Coast to Capital.
Willmott Dixon will work with ECE Architecture and Northgates to deliver the three-storey extension.
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Work is expected to complete by the end of 2020, with the college remaining open as usual for the duration of the works.
Roger Forsdyke, managing director at Willmott Dixon in the southern Home Counties, said: “This further builds on our recent track record that includes the Kennedy building for the University of Kent, another example of how we successfully work within a live campus site.
“We are really looking forward to working with the MET team on the next phase of their growth and, in the process, sustain regional companies and jobs with our ‘local pound’ commitment.”
Willmott Dixon has delivered over £350m of work to campuses at colleges and universities, including Birmingham, Swansea and Bournemouth, in the past three years.