Brother Laurence Hughes, head of the De La Salle Brothers in Great Britain and Ireland, told the Scottish Child Abuse Inquiry he would have expected other members of staff to have known what was going on.
Br Hughes said he had already met with survivors abused by the congregation in Northern Ireland and was prepared to do so in Scotland.
The inquiry, before Lady Smith, was told an estimated 9,300 pupils, many with emotional and behavioural difficulties, attended five institutions run by the Brothers between 1914 and 1992.
Last year 82-year-old Michael Murphy, known as Brother Benedict, was jailed for seven years for physically and sexually abusing boys in his care during the 1970s and 1980s at St Joseph’s School in Tranent, East Lothian.
He had previously been convicted for physically assaulting boys at St Ninian’s School at Gartmore, near Stirling, between 1960 and 1969 which was run by the De La Salle Brothers until 1982.
Asked by Colin MacAulay, lead counsel to the inquiry, if there was an acknowledgement abuse had taken place at St Joseph’s, Br Hughes said: “We acknowledge the abuse which was confined to the convictions against one brother.”
However, he admitted the abuse “may not be confined” to that particular case.
Lady Smith asked: “You accept other abuse happened?”
“I do,” replied Br Hughes, adding that he accepted failures had taken place.
He said: “Clearly the perpetrator is not going to be broadcasting what they’re doing, but there must have been somebody who was aware what was happening, whether that was a Brother,
a lay person or a member of the board...”
Br Hughes said as a chaplain elsewhere in the country he had personally heard allegations of abuse which he had passed to safeguarding officers and would have expected the same to have happened at St Joseph’s.
At Murphy’s trial in 2003, the court heard how the priest had tortured boys at St Ninian’s with electric shocks and had forced them to eat their own vomit. Murphy had also broken a child’s arm after he was cheeky to the priest.
Two lay members of staff at St Ninian’s, woodwork teacher Charles McKenna and groundsman James McKinstry, were convicted of a number of sex offences at the same trial.
Asked yesterday if he had a message for the victims, Br Hughes said: “When I became provincial (of the order) in 2015, it was towards the end of the Northern Ireland (child abuse) inquiry.
"My concern was certainly for the victims. One of the victims asked to speak to me and over the last two years I have spoken to at least four people who have been abused...”
He added: “I made it very clear that I am available.”
Asked if he was now making the same offer to victims in Scotland, he replied: “I am.”
Earlier, the inquiry heard from Sister Rosemary Kean, of the Good Shepherd Sisters, that her congregation had “no knowledge” of abuse taking place at five establishments for girls in Scotland.
It has previously been alleged that nuns physically assaulted children at the Good Shepherd Centre in Bishopton, Renfrewshire in the 1970s.
Sr Kean said: “We have no knowledge of abuse having taken place at any time and certainly no records. There were some allegations made against two sisters but they were acquitted of every charge.”