It also alleged that he concealed information when priests were reported to law enforcement and oversaw insufficient church investigations into abuse.
Abuse scandals have also shaken the Catholic Church in Australia, Chile and Ireland, among other countries.
Cardinal Donald Wuerl, the bishop of Pittsburgh between 1988 and 2006, was named in last month's scathing Pennsylvania grand-jury report as having concealed complaints of predator priests who sexually abused more than 1,000 children.
Some clergy from the Catholic Church have been carrying out abuse for decades.
Viganò said that Cardinal Wuerl was "well aware of the continuous abuses committed by Cardinal McCarrick and the sanctions imposed on him by Pope Benedict" and yet ignored the sanctions and allowed McCarrick "to reside at a seminary in Washington D.C".
Francis has received significant criticism over his handling of abuse, including an open letter from the Catholic Women's forum that has received more than 44,000 signatures.More news: Umpire changed the course of women’s final: Djokovic
Cardinal Daniel DiNardo, president of the US bishops' conference, asked for the meeting after Viganò last month accused the pope of knowing for years about sexual misconduct by former USA cardinal Theodore McCarrick and doing nothing about it.
The Pontiff will meet with top officials from the Catholic Church early next year, the Vatican press office confirmed this afternoon.
DiNardo has also said Vigano's accusations deserve answers.
The U.S. Conference of Catholic Bishops issued a policy in 2002 that is regarded as the gold standard policy.
In response to questions, she said there was no word yet on the expected release of the "possible and necessary clarifications" the council said were being formulated by the Holy See given the current debate on abuse in the church.
The commission set up by Francis said on Sunday that the fight against abuse must be a Church priority and emphasised the importance of listening to victims. Every sixth case involved rape and at least 1,670 clergy were involved, according to Spiegel Online and Die Zeit, which said they obtained the report that was due to be released September 25. Many cases were not brought to justice, and sometimes abuse suspects - primarily priests - were moved to other dioceses without new congregations being informed about the pastor's past, they said.