Kier, McLaren, Morgan Sindall and Wates are are expected to submit final bids for the job, which is due to start in September.
The new five-floor building extension will provide extra book storage, additional study spaces and a new lecture theatre and seminar rooms.
The rest of the library will be reglazed and modernised in an extensive refurbishment programme, which will get underway in 2014.
Landscaping and the provision of a new logistics/fire road will be required as part of the extension.
It is expected that the new building will achieve a BREEAM Excellent rating.
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