The Hoops confirmed on Monday the South Korea international was not travelling with the squad for their Champions League play-off in Helsingborg and was keen to move away from Parkhead.
And Jenkins revealed to STV that the Swans are targeting the signature of the 23-year-old in a deal thought to be worth £6million, saying: "We have agreed a fee with Celtic for Ki. We are currently holding discussions with player and agent over personal terms."
A deal has not yet been agreed and Jenkins is keen to get the transfer completed before he is usurped by any other potential suitors.
"We are aware of other clubs interest but we are hopeful we can conclude a deal in the next 24 hours. We feel he is a very exciting young player who has a good career in front of him," he said.
Hoops boss Neil Lennon, talking in Sweden after Celtic arrived in preparation for their match with Helsingborg, claimed the length of Ki's contract meant a potential windfall of £6million could not be ignored.
He said: "We are in advanced discussions with a club at the minute and we are hoping the deal will go through.
"I am not at liberty to say (if it's Swansea) but there may be interest from other clubs as well.
"He is not here, we have given him a bit of extra time because he has a lot to think about.
"He has only 15 months left on his contract and we feel it is a good offer and good business on our side."
Source: PA
Source: PA