The deal paves the way for preferred bidder Wates to start construction in July on the three planned buildings rising from 5 to 11-storeys.
This follows design changes including reducing floor to floor levels from 4.1m to 3.6m after bids for the construction project initially came in over budget.
The redesign process has put the project completion date back from 2018 to 2019, and still requires rubber stamping by planners.
The 170,000 sq ft project, which will neighbour the West Yorkshire Playhouse, will include science labs, computing and digital technology labs, dance, art, and film studios, and a performance theatre.
Plans also include a later 110,000 sq ft phase two, which has also been scaled down in height as part of the value-engineering process.
Suffolk-based Jackson Civil Engineering is the site enablement contractor.
Roger Marsh, Chair of the LEP said: “We’re delighted that Quarry Hill – one of the projects in our total £79m investment in FE infrastructure across the City Region and Leeds City College’s second major new city centre site – has begun to be realised.
“This transformational and ambitious development will give students from Leeds, Bradford and the wider City Region access to world class, state-of-the-art training facilities, allowing the College to closer align the powerhouse skills of our future workforce with the needs of our City Region businesses.”