The total cost of the scheme to renovate the library in Deansgate will be '15m.
The scheme will preserve priceless artefacts - the building holds one million items - and strip the institution of its stuffy academic image.
The library, part of Manchester University, has been awarded '8.19m by the Heritage Lottery Fund, and a bid has been made for a further '3m to the European Regional Development Fund.
But the university says public and charitable donations are also needed to realise the ambitious plans, which include adding a modern reception area, cafe and shop to the neo-Gothic building, as well as creating better access for disabled people.
The vision would also see relics currently under lock and key - including the St John Fragment, the oldest piece of the New Testament - on occasional display in new exhibition spaces.
It is hoped the improvements will triple the number of visitors to the library - claimed to be in the world's top 10 - bolstering the figure to over 100,000 a year.
Dr Stella Butler, head of special collections, said: "The future depends on this essential building work. Crumbling parts of the building need urgent attention.
"At the moment, we attract a very intellectual, academic audience and we want to make it a cultural centre which will welcome family groups."
The Duke of Devonshire, a former chancellor of the university, launched the appeal - called Unlocking The Rylands - urging people to "dig deep and help the library draw in couch potatoes".
A major portion of the cash will be used to replace the library's flat roof with a pitched one, completing architect Basil Champney's design for the building, which opened on New Year's Day, 1900.
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