South Korean investigators probing President Yoon Suk-yeol for alleged insurrection asked a Seoul court on Friday to extend his detention as the embattled leader again refused to be questioned.

On Wednesday, Yoon became the first sitting South Korean president to be arrested, in a criminal probe related to his short-lived declaration of martial law in early December. He is being held at the Seoul Detention Centre.

In order to hold Yoon in custody for longer, investigators at the Corruption Investigation Office for High-ranking Officials (CIO) need a court to approve a detention warrant for up to 20 days.

A CIO official told a briefing on Friday that investigators had made the request due to "the gravity of the crime". Yoon's current arrest term was due to expire by Friday evening.

The Seoul Central District Court struck down a challenge on Thursday from Yoon's lawyers over the legality of his arrest.

In a statement disseminated by his lawyer, Yoon said the jail cell where he is being held was "a little uncomfortable" but that he was doing well.

He also thanked his supporters, hundreds of whom gathered outside the prison in chilly conditions, saying: "I am grateful for the people's burning patriotism."

Yoon has stonewalled efforts by the CIO to interrogate him while his party has capitalised on political polarisation to improve its approval ratings since his arrest, polling showed.

"He has fully stated his basic position on the first day (of the arrest), and we believe there is no reason or need to answer the Q&A style back-and-forth," Yoon's lawyer, Seok Dong-hyeon, said in a statement.

The suspended president's legal team has denied Yoon masterminded an insurrection, a crime in South Korea punishable by life imprisonment or even, technically, the death penalty.

South Korea is grappling with its worst political crisis in decades, sparked by Yoon's attempt to impose martial law on December 3 that stunned the nation and was swiftly voted down by parliament.

Yoon was impeached on December 14 and he faces a Constitutional Court trial that started this week to decide whether to permanently suspend his powers or return him to office. (Reuters)