South Korean news outlets have been reporting for weeks that Ms. Choi, 60, who has known Ms. Park since the 1970s, is deeply involved in state affairs despite having no government job or background in policy. In a televised address last week, Ms. Park, whose office had denied the reports, acknowledged that Ms. Choi had edited some of her speeches and apologized to the country.

Ms. Choi’s father was the leader of a religious sect and a mentor to Ms. Park, and many South Koreans have come to believe that Ms. Choi also wields influence over the president. Ms. Park’s polling numbers have reached record lows, and thousands of people demonstrated during the weekend calling for her resignation or impeachment.

Earlier Wednesday, Ms. Park tried to recover from the scandal by replacing the prime minister, the No. 2 official in her government, as well as two other cabinet ministers. But the opposition parties that constitute a majority in Parliament said they would not approve the appointments. Opposition lawmakers said a more drastic government shake-up was needed to rebuild public trust, and they demanded that Ms. Park submit to questioning by prosecutors.

Prosecutors declined to say whether they planned to question the president.

The South Korean news media has said that Ms. Choi, in addition to coercing donations from businesses, helped her associates win lucrative government contracts. On Wednesday, in accusing her of attempted fraud, prosecutors said a private company Ms. Choi owns, The Blue K, had tried in vain to use phony business transactions to steal funds from one of her two nonprofit foundations and divert them to her company.

Ms. Park has long been accused of insufficient transparency in her government and of relying too heavily on a few trusted advisers, and the reports of Ms. Choi’s influence have fed into that criticism. But her family background is also a factor in the scandal.