Choi Soon-sil, who was detained on Monday, is accused of abuse of authority and attempted fraud.
Eight banks have also been raided in connection with the controversy, South Korean media reported.
The scandal has prompted widespread anger and intense public scrutiny.
What's the scandal about?
At the heart of the scandal is the influence that Choi Soon-sil, the daughter of a shadowy religious cult leader, may have had on South Korea's President Park Geun-hye.
What is Ms Choi accused of?
The allegation is that Ms Choi was involved in high-level decision making and handled classified documents without security clearance. She is also accused of embezzlement - allegations she has denied.
Why are they so close?
The true extent of Ms Choi's influence is unclear. The women have been friends since the 1970s when Ms Choi's father, Choi Tae-min, befriended Ms Park's family. He eventually became Ms Park's mentor.
What has the president said?
She is embroiled in a battle for her political life as a result of the scandal. She has apologised to the public for causing national concern.
After two days of questioning, prosecutors filed the formal request for an arrest warrant to enable them to detain her longer for questioning. Seoul Central District Court will review the request on Thursday, a court spokesman said.
South Korea's president has responded to the crisis by reshuffling her government.
Kim Byong-joon, a senior presidential secretary under a previous administration, was named as prime minister to replace Hwang Kyo-ahn. The role of prime minister is largely symbolic in South Korea, where power is concentrated in the presidency.
Yim Jong-yong, currently Financial Services Commission chairman, was named as the new finance minister and deputy prime minister, replacing Yoo Il-ho.
A new minister of public safety and security has also been appointed.
But the opposition says the reshuffle is an attempt to deflect attention from Ms Choi.
Surveys show Ms Park's approval rating is about 10% and about half of respondents say she should resign or face impeachment.