The city has only played host to two Ireland senior men's internationals
Turner's Cross is the unlikely frontrunner to host Ireland's last game ahead of Euro 2016.
Martin O'Neill's side face Switzerland and Slovakia in the coming days in the Aviva Stadium, before a game against the Netherlands on May 27th.
The Irish Daily Mirror have reported that Belarus may play Ireland in the home of Cork City on May 31st. Ireland are using the nearby Fota Island Resort as a training base from May 30th to June 3rd.
The last time Ireland played a home game outside Dublin was against South Africa in Thomond Park, Limerick in 2009. Picture by: Owen Humphreys / PA Archive/Press Association Images.
If the game goes ahead in Turner's Cross it will only be the third time the city has hosted a senior men's international match and will be the first in the venue. In 1939, Ireland drew 2-2 with Hungary in The Mardyke.
As part of the Cork 800 celebrations in 1985, Eoin Hand's side drew 0-0 with Spain in Flower Lodge. That ground has since become Pairc ui Rinn.
Belarus play Northern Ireland in Belfast on May 27th and may be prepared to travel south to Cork to play their second friendly in four days.