The library, designed by Giles Gilbert Scott whose other designs include the red telephone box and Battersea Power Station, is seeking entries that are an "innovative interpretation of the site, its context, use and history - as well as their ability to integrate contemporary ecological research".
Judges include designer Tom Stuart-Smith and TV historian Julia Beard.
Ingrid Schroder, a design fellow from the university's Department of Architecture, said: "We're looking to prompt a true diversity of responses from anyone who'd like to enter. This is an ideas competition and that's exactly what we want to see from people."
The competition is also acting as a forum for urban design issues. For more information, see www.cambridgelibrarylandscape.com.