SEOUL (Reuters) - South Korean President Park Geun-hye's approval rating remained at an all-time low for a second week amid a deepening political crisis, according to a Gallup Korea poll released on Friday, underscoring the challenges faced by her administration.

Park's presidency has been rocked by allegations that a personal friend of hers used her ties to meddle in state affairs and wield improper influence.

Gallup said its survey of 1,003 South Koreans from Nov. 8-10 showed that 90 percent of participants disapproved of her performance, up from 89 percent last week and a new high.

Park's support dipped to single digits across the country, including in her traditional support base in the conservative stronghold of the southeast. She had zero support among people under the age of 30, the poll showed.

The results of the Gallup poll came after Park gave her second public apology last week and indicated she was willing to relinquish some power.

Park offered to withdraw her nominee for prime minister this week if parliament recommended a candidate that opposition parties could agree upon, but her offer did little to quell public anger.

The post of prime minister is largely a figurehead role in South Korea, where power is concentrated in the presidential office.

The two main opposition parties have rejected Park's offer and demanded she hand over most of her presidential authority to a new prime minister and the Cabinet.

Some lawmakers have called on her to step down but the main opposition parties have not raised the idea of launching impeachment proceedings.

A large crowd was expected at an anti-Park rally on Saturday, with organizers saying the number could reach 1 million, according to Korean media reports.

(Reporting by Ju-min Park and Jack Kim; Writing by Se Young Lee; Editing by Paul Tait)