The call in the House of Lords for action came after a report found monks at Ampleforth in North Yorkshire and Downside in Somerset hid allegations of "appalling sexual abuse" against pupils as young as seven to protect the church's reputation.
The Independent Inquiry into Child Sexual Abuse (IICSA) also said the abuse spanning four decades was likely to be "considerably" more widespread than previously thought.
It made the claims in a scathing report on the English Benedictine Congregation, which has 10 monasteries in England and Wales.
Ampleforth and Downside are two schools linked to the monasteries, run at times by "secretive, evasive and suspicious" church officials who avoided reporting misconduct to police and social services.
Allegations stretching back to the 1960s encompassed "a wide spectrum of physical abuse, much of which had sadistic and sexual overtones", according to the report.
Ten individuals linked to the schools, mainly monks, have been cautioned or convicted over sexual activity or pornography offences involving a "large number of children".
Raising the scandal in the Lords, Liberal Democrat peer Baroness Walmsley said: "Is the Minister aware that the committee had evidence that one of the schools consulted its legal adviser as to whether it was legally obliged to report the abuse that it knew about. Having learned that it was not so obliged, it decided to cover it up.
"How much more evidence do the Government require of the need for mandatory reporting of child abuse in regulated activity? Will the Government now follow the evidence and respond with legislation?"
Education Minister Lord Agnew of Oulton said: "It is absolutely unacceptable for anyone to conceal abuse. The Government are committed to ensuring that legislation can adequately deal with this."
He added: "What individuals and organisations should do is already clear in statutory guidance."
Labour frontbencher Lord Watson of Invergowrie pointed out the Independent Schools Inspectorate had given both schools "a clean bill of health when it comes to safeguarding".
He asked what steps were being taken to "ascertain how the inspectorate managed to miss the continuing failings at these wretched establishments".
Liberal Demcorat peer Baroness Brinton said: "The Minister has talked about the last chance saloon and said that schools know what they should be doing. However, they are still not reporting all cases.
"When will the Government introduce mandatory reporting for regulated activities?"
Lord Agnew said: "I know that there are calls for mandatory reporting."
But highlighting a consultation carried out he added: "We had 760 responses from social workers, police officers and other connected parties.
"Some 70% of them felt that mandatory reporting would have an adverse impact; 85% said that it would not, in itself, lead to the appropriate action being taken.
"However, over the last few years we have prioritised sexual abuse as a national threat, to empower police forces to maximise skills and expertise."
Stay up to date with the latest news stories from a Christian perspective. Sign up to our daily newsletter and receive more stories like this straight to your inbox every morning.
