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The Turner Contemporary could be expanded at a cost of £6 million - with half of the funds coming from Kent County Council.
Although in the early stages, the proposals include increasing the exhibition space and improving the education area, café and shop.
Access would be improved and more toilets put into the building on Margate seafront.
Turner says the building will be reconfigured to allow the gallery to connect with more people, increase cultural tourism and generate vital income.
A spokesman said: "Additional space created will be designed to make the building more sustainable, improve our visitor experience and expand the potential of our work which will enable us to become more financially resilient in the future."
According to a report by KCC last year, the gallery has outgrown its existing buildings.
The proposed scheme, which would be known as Turner Contemporary Phase 2, will cost £6m - £3m from Arts Council England if the funding bid is successful and a further £3m of match funding from KCC.
County councillor Mike Hill, who has been involved with the gallery since its inception, says the reconfiguration is a bold and brave concept.
“It’s early days and these are the first plans to see what people think and we’re very pleased by the public response,” he said.
“We’ve been a victim of our own success. We’ve had many more visitors than we expected and the gallery is not big enough for the reviewed number of visitors.
“The large number– we’ve had well over two million – has led to some wear and tear and it needs some brightening up and work to bring it back up to a first class standard.”
The phase 2 scheme, once decided, would not begin until after the gallery hosts the Turner Prize next year.
Cllr Hill added: “We couldn’t sensibly start work until after 2019.
"We’re aiming to hit the ground running as soon as that has finished.”
Any expansion will be subject to funding and planning permission.