Chung Yoo-Ra, the 20-year-old daughter of the woman dubbed South Korea's "Rasputin", is one of the figures in the influence-peddling scandal that sparked massive street protests demanding the removal of President Park Geun-Hye.
Chung was held on Sunday for overstaying her visa, after South Korean authorities issued a warrant for her arrest.
"The processing of the extradition case against the 20-year-old woman, Chung Yoo-Ra, can now begin," Danish prosecutors said in a statement, adding South Korea had emailed an extradition warrant.
The equestrian, who has reportedly bought horses and trained in Denmark in the past, has told police that she was in the country due to her involvement in the sport.
Chung denied any wrongdoing at a hearing on Monday, where the court ordered that she be detained until January 30.
"It may be that we will be able to reach a decision in the case before 30 January, but of course it would be natural to request an extension of the remand if this is not the case," said Mohammad Ahsan, deputy director of the public prosecutor's office.
Choi, a secret confidante of Park, is accused of using her ties with the president to force top firms including Samsung to "donate" nearly $70 million (67 million euros) to non-profit foundations which Choi then used as her personal ATMs.
She is also accused of using her influence to secure her daughter's admission to an elite Seoul university, with a state probe revealing the school had admitted Chung at the expense of other candidates with better