The Episcopal Parish Church of Saint John, Haj Yousif, in Khartoum was reportedly demolished on 18 June on the orders of Sudanese government officials.
The WCC and AACC have denounced the action as the latest in a series of "calculated attacks" on minority communities and Christians in particular.
On 21 April, the Sudan Evangelical Church Bible School in Khartoum was destroyed and books including the Bible set on fire in full view of the police.
Two days later, security forces occupied the premises of the Sudan Council of Churches and Sudan Aid in Nyala, Darfur, and confiscated property.
The WCC and AACC warned that Christians of Muslim background were being targetted and dispossessed of their property and their spouses.
In a joint statement, the general secretary of the WCC, Olav Fykse Tveit, and the general secretary of the AACC, Dr André Karamaga called for an investigation into the actions of the authorities and appealed to Christians to pray for faith communities in Sudan.
"We once again regret that despite repeated rhetoric about freedom of religion and the protection of the minorities in the Republic of Sudan, the government policy seems to be bent on threatening and discriminating against Christians in Khartoum," they said.
"By protecting religious fundamentalists who wreck mayhem and havoc on innocent civilians with impunity, the Republic of Sudan undermines the tenets on which a multi-religious society is based.
"The ecumenical movement cannot remain silent while such a horrific violation of human rights and threat to lives continues unabated. Indeed, as the Church, we are called to promote and assure abundant life, a possibility that seems to be fading for many Christians in Sudan."