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.
Yoon Suk-yeol's supporters have been gathering outside the prison where he is being detained. (AP PHOTO)
Investigators at the Corruption Investigation Office for High-ranking Officials (CIO) need a court to approve a detention warrant for up to 20 days to hold Yoon in custody for longer.
A CIO official told a briefing on Friday that investigators had made the request due to "the gravity of the crime".
Seoul Western District Court will review the request on Saturday afternoon, Yonhap News Agency reported.
In a statement disseminated by his lawyer, Yoon said the jail cell where he was being held was "a little uncomfortable" but that he was doing well.
He also thanked his supporters, hundreds of who gathered outside the prison in chilly conditions, saying: "I am grateful for the people's burning patriotism."
Yoon has stonewalled efforts by the investigation office to interrogate him while his party has capitalised on political polarisation to improve its approval ratings since his arrest, polling shows.
"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.
The Seoul Central District Court struck down a challenge on Thursday from Yoon's lawyers over the legality of his arrest.
South Korea is grappling with its worst political crisis in decades, sparked by Yoon's attempt to impose martial law on December 3 which stunned the nation and was swiftly voted down by parliament.
Yoon was impeached on December 14 and faces a Constitutional Court trial that started this week to decide whether to permanently suspend his powers or return him to office.
Lawyer Seok Dong-hyeon says Yoon Suk-yeol has "fully stated his basic position". (AP PHOTO)
Opinion polls have shown most South Koreans support his impeachment, but Yoon's legal plight and defiance over his arrest appear to have fired up some of his supporters.
The approval rating for Yoon's ruling People Power Party (PPP) was 39 per cent in a Gallup Korea poll released on Friday, up from 34 per cent a week ago and overtaking the main opposition Democratic Party on 36 per cent for the first time since August.
Gallup Korea said Yoon and his party's constant messaging toward supporters appeared to have had an impact when political divisions deepened around his arrest.
Supporters gathered outside the jail on Friday welcomed the uptick in the party's approval rating.
"People are now waking up to the fact that our country is at risk," said Noh Eun-sook, 57, who was wearing a woolly hat and chanting for Yoon's release.
Yoon was arrested after a weeks-long stand-off when more than 3,000 police officers swooped on his residence.
A previous attempt to arrest him on January 3 failed after Yoon's personal security blocked investigators.
Kim Sung-hoon, acting chief of the Presidential Security Service, was arrested on Friday for thwarting investigators' initial attempt, Yonhap said.
Speaking to reporters, Kim said he was carrying out his "legitimate security duties" and denied allegations by some opposition MPs that Yoon ordered guards to use weapons against investigators trying to arrest him.
While key ally Washington has criticised Yoon's declaration of martial law, US national security adviser Jake Sullivan warned last week there was a risk that nuclear-armed North Korea could try to exploit the political situation in South Korea.
North Korea has largely avoided public comment on the situation in Seoul, but Yoon's arrest was reported in state media on Friday, two days after the event.
