A sports arena is to be transformed into one of the UK's biggest museums under plans involving Glasgow University and the city council.

The plan will see the University's world-famous Hunterian Museum and Gallery moving to the Kelvin Hall, creating a museum hub with the nearby Kelvingrove Museum and Art Gallery.

In conjunction with Glasgow City Council, which owns the Kelvin Hall, the university is planning to make the facility home to its 1.3 million-object collection, as well as a study centre, huge public art gallery and sports facility.

With the Kelvingrove Museum and Art Gallery across the road, and the Riverside Museum nearby, the council believes the west end could rival South Kensington in London – home of the V&A, the Science Museum and the Natural History Museum – as a cultural hub for tourism and academia.

The new museum, gallery and store is to be created in two phases, at a cost of about £60 million. The first phase – transforming a large part of the building into a joint collections and study centre – is planned to be completed by the end of 2016, with the aid of a grant from the Heritage Lottery Fund.

The second stage, creating new public galleries and special exhibition and education spaces, would be completed by 2020.

In this phase, the current Hunterian Art Gallery will close and move to the Kelvin Hall, which will be re-vamped with the input of leading architects Page/Park.

Professor David Gaimster, director of the Hunterian Museum and Gallery, wants to quadruple the percentage of items from the university collection on public display, from the current 0.5% to 2%.

He said: "By the end of the decade, we hope there will be a new Hunterian, at Kelvin Hall, and that branding at the front of the facade will be the Hunterian – a new Hunterian and new museum for Scotland. That is the vision.

"The 'do nothing' option is not viable for the Hunterian, because we are constrained by the very small gallery footprint that we have. We cannot do the big major exhibitions we would like to do; we cannot provide the right educational projects we would like to do."

The joint university and council facility will help house the majority of the Hunterian collection, built on Dr William Hunter's founding bequest, and first opened in 1807.

It would also contain about 400,000 items from the City of Glasgow's collections that are currently not on display, including the only surviving complete suite of interiors by Charles Rennie Mackintosh, for the Ingram Street tearooms.

Mr Gaimster said: "It will be storage and study, it will be open-access; there will be tours, and events – all kinds of stuff we cannot do at the moment.

"Phase One will also include a sports development, so it will be multi-purpose, and once that foothold has been created, then we have the option to develop the new gallery space, exhibition and education space."

The second phase would cost about £35m and turn the B-listed building, with its 171,000sq feet hall, into one of the premiere cultural venues in the UK, especially for travelling art exhibitions.

Mr Gaimster added: "The city will be working with the university for the first time on this capital development project, a unique venture, and develop Kelvin Hall as a culture space – the biggest in Scotland.

"It's a long time in the planning. It's a vast site, with the biggest roof span in Scotland, and it can only be developed in this phased approach – to do it in one go would cost £150m or so."

A spokesman for Glasgow Life, the council agency that runs the Kelvin Hall, said: "This is a plan that would secure the future for one of the city's most important buildings."

It will be storage and study, there will be tours and events ... all kinds of stuff we cannot do at the moment