Park Geun-hye has come under attack over the actions of her old friend and mentor Choi Soon-sil.
On Monday, Ms Choi was detained by police, accused of influence peddling and interfering in state affairs.
Eight banks have also been raided in connection with the scandal, South Korean media reported.
Critics say Ms Choi used her closeness to the president to solicit corporate money for foundations she ran, as well as embezzling state funds and directing presidential decisions, despite a lack of office, expertise or security clearance.
The scandal has prompted widespread anger.
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 in the reshuffle.