The aim is to create 200 en-suite bedrooms at Pantycelyn and public spaces for community and student use.
There were student protests when closure of the Grade II halls was proposed, with campaigners accusing the university of not placing enough importance on the Welsh language.
If planning is approved, the halls should reopen by September 2019.
Gwerfyl Pierce Jones, chair of the Pantycelyn project board, said: "This is another important milestone in the history of Pantycelyn as we move a step closer again to reopening the building as a hall of residence."
Plans will be on display at the Royal Welsh Show and at the National Eisteddfod, and can be viewed on the university's website.
Rhodri Llwyd Morgan, pro vice chancellor at Aberystwyth University, said: "Thousands of students have fond memories of having lived in Pantycelyn over the years and we are keen now for these plans to be seen by as many as possible.
"As the application for planning permission goes before the county council, the university is continuing the work of ensuring the necessary funding is in place for this ambitious project."
Aberystwyth University has submitted an application to planners to redevelop its Welsh language halls.