South Korean media reported on Monday that the pair suffered bruises in the crash on Saturday.
Segye Ilbo newspaper, citing police, said Jungkook violated unspecified traffic rules while driving his Mercedes Benz and hit the taxi on a Seoul street.
Jungkook, 22, admitted that he caused the accident by violating traffic laws and reached a settlement with the driver, according to BTS' agents Big Hit.
BTS: The world's biggest boyband - In pictures
A statement said the pair both avoided serious injury and added: "We apologise again to the victim and also to fans for causing concern."
Seoul police said Jungkook was booked and an investigation of his case was underway, in line with traffic law.
Yonhap news agency cited police as saying that they plan to summon Jungkook soon.
A police officer in charge of traffic accidents in Seoul, who requested anonymity because the investigation was underway, described the case as minor because the incident did not cause much human or property damage.
They also said the case did not involve drunken driving or any other serious offences.
BTS has a large international following and is made up of seven members.
It was the first K-pop act to debut atop the Billboard Album chart last year with "Love Yourself: Tear."
Additional reporting by agencies.
