The Swansea Labour-controlled Council cabinet will be asked to award the contract for YGG Tan-y-lan at its meeting on Thursday, October 17.
Work will be scheduled to start on the new build early in 2020 with the new school ready to open in autumn 2021.
YGG Tan-y-lan is currently based at the former Graig Infant School site in Morriston.
A new spacious school with first class facilities will be built at Beacons View Road in Clase and once completed it will have capacity for 420 pupils plus nursery to help meet the growing demand for education through the medium of Welsh in the area.
The £9.9m investment is being jointly funded by Swansea Council and the Welsh Government as part of the 21st Century Schools and Colleges programme.
Labour Cllr Jennifer Raynor, Swansea Council’s Cabinet Member for Education Improvement, Learning and Skills, said: “The awarding of the contract is the next milestone for this project.
“The new school will be large enough to accommodate two form entry so it will be able to cater for the increased number of pupils being taught through the medium of Welsh.
“It will have a mini all-weather pitch and multi-use games area which will be available for community use outside of school hours.
“There will also be childcare on site to allow Welsh-medium numbers and interest to grow as the school grows.”
As with all building projects commissioned by Swansea Council the contractor will be expected to use local labour and suppliers where possible and provide apprenticeships and work experience opportunities to support the local economy.
Work is already progressing well on two other brand new school builds in Swansea.
A new home for the pupil referral unit is taking shape in Cockett and will replace outdated accommodation at different locations throughout the city bring the service for some of Swansea’s most vulnerable young people together on one site.
Pupils and staff at Gorseinon Primary School are looking forward to the completion of their new school that will replace the outdated Victorian buildings they currently use.
Another new build is set to start next year for a new home for YGG Tirdeunaw on land next to Ysgol Gyfun Gymraeg Bryn Tawe.
Cllr Raynor added: “The council’s Band B programme which will see almost £150m invested in school infrastructure over the next few years is now really taking shape.
“Along with our colleagues in Welsh Government we are determined to provide improved surroundings with modern facilities to help our young people reach their full potential.”