The new state-of-the-art buildings would be built on land next to Ysgol Gyfun Gymraeg Bryn Tawe.
It's currently based at the former Daniel James secondary school site at Heol Ddu.
The existing school currently provides 420 full-time places and 60 nursery places but the buildings are no longer suitable for the increasing demand for places at the school.
The full planning application seeks permission for a new build that would provide places for 525 full-time pupils and 75 nursery places along with a Flying Start/Dechrau Deg’ initiative.
An extensive consultation exercise was carried out with pupils, parents, staff and the wider community prior to Christmas and a statutory notice was published in January stating the council’s intention to increase the capacity at YGG Tirdeunaw.
Should the plans be approved the new school site will be jointly funded by Swansea Council and the Welsh Government under the council’s 21st Century School Programme.
The Welsh Government has set a target of growing the number of Welsh speakers to one million by the year 2050.
Swansea’s Welsh In Education Strategic Plan, which was endorsed by ministers last year, stated that an additional 850 places could be created at Welsh-medium schools in the city to meet growing demand.
Cllr Jennifer Raynor, Swansea Council’s Cabinet Member for Education Improvement, Learning and Skills, said: “The number of pupils being taught through the medium of Welsh in our schools has been increasing for a number of years.
“The council is committed to meeting this growing demand and I was very pleased that a planning application for a brand new school for YGG Tan-y-lan at a new site in Clase was approved by the council’s planning committee earlier this month.
“Should our application for a new school for YGG Tirdeunaw also be successful then when both projects are completed there will be an additional 409 places at Welsh medium primary schools in Swansea along with improved nursery and Flying Start provision.
“As part of our budget, which was approved in February, the council has plans to invest almost £150m in both English and Welsh medium schools in the coming years to ensure that our children and teachers have access to the first class facilities that they deserve.”