San Diego's first-year general manager A.J. Preller met at least twice at the Winter Meetings with Kim's agent, Melvin Roman.
In November, the Padres won the bidding rights to Kim with their $2 million posting fee to his team, SK Wyverns, who were, according to reports, disappointed and were expecting an offer in the range of $10 million.
But Kim's team ultimately accepted the bid, signaling the Padres had 30 days to reach a deal with him.
The Padres recently had Kim in San Diego, where he took a physical and met with team officials, and even Preller sounded optimistic this week that a deal could be reached.
Because it wasn't, the Padres won't have to pay his posting fee. Kim is free to return to the Korean Baseball Organization.
In 2014, Kim finished 13-9 with a 3.42 ERA in 28 starts. After winning the KBO MVP Award in '08, Kim struggled a bit with shoulder and wrist injuries that cost him most of the 2011-13 seasons.
Corey Brock is a reporter for MLB.com. Keep track of @FollowThePadres on Twitter. This story was not subject to the approval of Major League Baseball or its clubs.