The newspaper, without citing a source, alleged that former Fifa vice-president Chung tried in 2010 to influence the vote in his home country's favour in violation of the body's code of ethics.
In a letter dated October 2010, Chung made an offer to Fifa's executive committee to create a fund to support various football-related projects to the value of $777 million.
"Chung matched his commitment with a condition -- that the World Cup was awarded to South Korea," Welt am Sonntag reported.
Such moves could be deemed to be in breach of the Fifa code and the newspaper claimed that Germany's Hans-Joachim Eckert, who heads the body's ethics committee, could rule on the allegations.
The 2022 World Cup was finally awarded in December 2010 to Qatar.
Chung, 63, who served on the executive committee until 2011, sent two letters to Sepp Blatter, the outgoing president of the organization, in order to stop the investigations concerning him, but without success, added Welt am Sonntag.
Contacted by AFP, Fifa were not available for comment.
Chung, a member of the family that owns the various arms of the Hyundai conglomerate, launched his campaign to succeed Blatter as Fifa president on Monday.
World football's governing body has been embroiled in a corruption scandal since seven Fifa officials were arrested in a Zurich hotel ahead of their congress in May.
The seven are among 14 people facing US charges over more than $150 million in bribes allegedly paid for television and marketing deals.
Others under suspicion
Swiss investigators are also probing Nicolas Leoz, a veteran Fifa official in South America, a Swiss newspaper said on Saturday.
Leoz, an 86-year-old Paraguayan, headed the regional football organization Conmebol from 1986 to 2013 and was a member of Fifa's executive committee for 15 years.
Swiss officials are also investigating Eduardo Deluca, a long-term general secretary of Conmebol, after private bank Pictet tipped them off in June about suspicious movements in a bank account, the Zurich daily Tages Anzeiger said.
Millions of Swiss francs were paid into the account by Fifa and by Japanese advertising agencies, and much of the money was then transferred to Paraguay, it said.
Suspicions are also being directed at an account with the Credit Suisse bank, it said.
The Swiss prosecutor's office declined to comment on the report.
Leoz's attorney said his client was unaware of any inquiry while Deluca could not be contacted, the paper said.