The retired teacher, 83, admitted to the abuse in the Capital, where he taught at Fettes College and Edinburgh Academy in the 1970s, in court documents seen by the BBC.
Mr Campbell is one of a number of former pupils who have made allegations about the man.
The broadcaster earlier spoke about witnessing sexual abuse of a fellow pupil at Edinburgh Academy.
It has now emerged the teacher, who has not been named, is fighting an extradition to Scotland to face charges, with the UK extradition request approved by South Africa in 2020.
According to the BBC, the South African court documents reveal the original warrant for his extradition related to seven Scottish charges - six of lewd, indecent and libidinous practices and behaviour and one of indecent assault.
The teacher has lodged an appeal against the extradition, with a hearing due in October.
He admitted that the charges against him were serious but said: "I feel absolute remorse for what I have done to the boys."
The man said it would be "unjust, unreasonable and too severe a punishment" to be extradited to Scotland.
According to the BBC, in documents linked to the extradition proceedings, the man said that while teaching at Edinburgh Academy he had "urges to touch the students inappropriately and on occasion I did so".
The teacher also admitted he had urges to touch pupils in South Africa, where he taught at two primary schools in Cape Town in the 1960s.
Court documents also reveal that he decided to travel to the UK in 1967 for psychiatric treatment for his "problem".
After three months at the Royal Edinburgh Hospital, he went on to obtain a diploma in education at Moray House College the following year before taking up a post at Edinburgh Academy.
Married in 1973, he was hired as a primary school teacher at Fettes Junior College where he said his inappropriate behaviour continued, according to the papers seen by a BBC reporter.
He said that he returned to hospital for further treatment "for my drinking and inappropriate behaviour".
In 1979, following a complaint from a pupil, he said he admitted the behaviour and agreed to leave the college, before returning to South Africa with his wife. He taught at a primary school for 27 years and claimed there were no further incidents of inappropriate behaviour.
Last week Nicky Campbell, told listeners to his podcast and BBC Radio 5 Live show that he witnessed a man sexually assault a pupil at Edinburgh Academy.
He also spoke about being beaten up by another teacher.
Edinburgh Academy has previously apologised to anyone who suffered abuse and insists it now has robust safeguarding measures in place.
Fettes previously offered a "full and unreserved apology" to those who suffered abuse while at the school.