The company is in talks with Leicester but is not in a position to reveal further details, the Bangkok Post newspaper quoted Voramas Raksriaksorn, the company's communications head and daughter of its owner, as saying.
The deal is expected to be worth around $2 billion baht ($62.4 million), Thai television reported. A company spokeswoman could neither confirm or deny the reports.
Leicester have won the League Cup three times and have spent much of their 126-year history moving between the top two tiers of English football.
They have played three seasons in the Premier League since its inception in 1993, the last in 2003/04.
King Power owns 76 outlets at domestic terminals and has a duty-free monopoly at Bangkok's Suvarnabhumi international airport, one of Asia's largest.
On the club's website Leicester confirmed takeover negotiations were ongoing but did not name the prospective buyers, who they said would make "excellent partners to take the club forward."
The last Thai venture into English football ended in acrimony when former billionaire Premier Thaksin Shinawatra, ousted in a 2006 coup and still living in exile, bought Manchester City and sold the club a year later having angered fans with his treatment of manager Sven-Goran Eriksson.
