The proposals, which also include the restoration and extension of Templemore Baths, are part of Belfast City Council's £105 million Leisure Transformation Programme.
The redevelopment of both sites represents a major part of a £40m investment in east Belfast's leisure facilities.
A public consultation has opened and will continue until Sunday, 14 January.
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The Heritage Lottery Fund is contributing £5m towards the £17m investment at Templemore Baths where the focus will be on heritage preservation, spa provision and increased swimming provision.
Chair of the Strategic Policy and Resources Committee, Councillor Deirdre Hargey, said: "I would urge everyone to get involved in the public consultation – these are major investments and we want to ensure that they accurately reflect the needs and wants of the people of the area.
"The facilities that will be available at Avoniel and Templemore, once work is complete, along with those on offer at the new Robinson Centre when it opens its doors in 2019, will bring a wide leisure choice to the people of east Belfast.
"In total we're developing seven new centres across the city through our Leisure Transformation Programme. Each centre will provide state of the art leisure facilities while also offering their own unique facilities."
(CD/LM)
