Ki, who recently broke Park Ji-sung's record for the most Premier League appearances by a Korean player, has been at the Liberty Stadium for six seasons since joining from Celtic in the summer of 2012.
The 29-year-old has been one of Swansea's most consistent performers during their recent rejuvenation but has not signed a contract extension at the club and is expected to leave for free in the summer.
He was voted Swansea's player of the season in 2014-15 after scoring eight goals to help the Swans to 8th in the Premier League, but he has struggled to replicate that form since, scoring only once this season - a late winner against Burnley two weeks ago.
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Milan reportedly want Ki as an alternative to Franck Kessie, the Ivorian midfielder who has missed only one Serie A game for the Rossoneri this season. Milan have been much improved under Gennaro Gattuso, but the Italy legend clearly feels that he needs to strengthen his options ahead of next season.
Ki is the current captain of the South Korean national team and will lead his country at the 2018 World Cup, where they face Germany, Sweden and Mexico in Group F.