Ki's future at Celtic has been the subject of conjecture this summer with a number of clubs credited with an interest in the midfielder.
The 23-year-old has impressed at Parkhead since joining Celtic from FC Seoul during the January transfer window in 2010, helping them win the SPL title last season, while also catching the eye at the Olympics with South Korea.
The likes of Arsenal and Queens Park Rangers have been linked with a move for Ki, but it appears Swansea are in pole position to land the South Korean.
Swansea boss Michael Laudrup is in the market to bolster his midfield options after selling Joe Allen to Liverpool earlier this summer.
Celtic are thought to be willing to sell Ki for the right price and talks are thought to be at an advanced stage between both clubs.