Bishop of Buckingham, Rt Revd Dr Alan Wilson, leads more than 100 senior Anglicans in a letter to the two leaders as they prepare for a meeting of 38 primates from the global Anglican Communion.
The meeting is expected to focus on the treatment and acceptance of gay people within the Church.
The Church must accept LGBT people have been treated as "second-class citizens", the letter says,
It says: "We understand that the primates come from a variety of contexts with differing ways of interpreting the scriptures, but we urge you to be prophetic in your action."
Justin Welby and John Sentamu are urged to take an "unequivocal message" to the other leaders.
The pair are told to accept that the Church has failed in its "duty of care" to homosexuals and must apologise for "our part in perpetuating rather than challenging ill-informed beliefs".
Bishop Alan told Premier: "There really is a time now for action, not just for saying sorry and saying we got it wrong and whatever we say about you we really like you, really that's not good enough.
"We've got to take action to stop some of the dreadful things that are going on around the world aimed at LGBTI people."
Premier understands senior Church officials are worried leaders from African nations could walk out on the talks over the treatment of gay people.
They're thought to be deeply unhappy at the acceptance of gay Christians in countries like the United States and feel the Church can no longer stay as one body with such differing views.
Those who believe homosexuality is a sin cite Bible verses in support of their position.
In July Revd Will Pearson-Gee, the Rector of Buckingham, told the Anglican group Reform: "The Bishop of Buckingham courts publicity for his revisionist agenda and gets it.
"He has sadly become a figure of disunity in the Oxford Diocese and a cause of grief to many faithful Anglican Christians.
"The version of marriage he espouses is incompatible with Biblical Christianity."
Susie Leafe, Director of Anglican evangelical campaigning network Reform, told Premier the letter was not a surprise: "Knowing some of the things that those who have signed it have said in the past, I think there's a real confusion over what it means to follow Jesus as our Lord, to accept him as our saviour."
Bishop of Buckingham, Rt Revd Dr Alan Wilson, speaking to Premier's Antony Bushfield:
Susie Leafe speaking to Premier's Antony Bushfield: