Vigano gave the bombshell statement to conservative Roman Catholic media outlets during the pope's visit to Ireland, which was dominated by the Church's sexual abuse in that country and others around the world.
The letter can be read here. Francis was so enraged that Davis' supporters had leaked word of the meeting that the Vatican subsequently insisted he only held one private audience while there: with one of his former students, a gay man and his partner. He answered questions on abortion, sexuality and the child abuse crisis in the Church - but then someone asked him about a letter that accuses Francis himself of covering something up.
Francis was asked by a USA reporter during an airborne press conference Sunday if Vigano's claims that the two discussed the McCarrick allegations in 2013 were true.
He said Francis asked him about McCarrick when they met on June 23, 2013, at the Vatican's Santa Marta hotel where the pope lives, three months after Francis was elected pope.
"You have the sufficient journalistic capacity to draw conclusions", the pontiff told reporters in the Vatican's plane.
Archbishop Vigano said in the letter that he had told Pope Francis of allegations against Cardinal McCarrick in June 2013 shortly after his election as Pope by his fellow cardinals.
He says the pontiff knew about allegations of sex abuse by a prominent USA cardinal for five years before accepting his resignation last month.
Cardinal DiNardo's statement renews the cardinal's earlier invitation to the Vatican to proceed with an apostolic visitation to the United States "to seek the truth".
David Luiz exclusive: hard times at Chelsea taught me a lot
It's great to understand football - it never stops - since I started playing football I always learn from every coach, it's great. "Then, if I enjoy the game, maybe the supporters enjoy the game".
The National Catholic Register and another conservative site, LifeSiteNews, published the letter attributed to Archbishop Carlo Maria Vigano on Sunday as the pope was wrapping up a two-day visit to Ireland.
In a July 29 statement, Cardinal Wuerl said: "When the first claim against Archbishop McCarrick was filed in the Archdiocese of NY, the Archdiocese of Washington reviewed its own files and found no complaints of any kind made against Archbishop McCarrick".
He had "begged for God's forgiveness" for past clerical abuse scandals during his landmark trip to Ireland, where they have badly damaged the image of the Catholic Church.
Nicolas Seneze, the Rome correspondent for the French daily La Croix, echoed that there is "a clear desire to attack Francis", telling AFP that "those who regard Francis as unsafe will stop at nothing".
Pope Francis used the penitential rite of the Mass to list a litany of different types of abuse and mistreatment inflicted on Irish people by Church figures, and the cover-ups of sex crimes. We ask for forgiveness.
The Irish Catholic Church has been rocked by revelations of paedophile priests, sexual abuse in Catholic-run orphanages, and the exploitation of women in mother-and-baby homes.
DiNardo said he was "eager" to meet the pope "to earn his support for our plan of action" that would make reporting of abuse and misconduct by bishops easier, and improve procedures for resolving complaints against bishops.
Pope Francis heard Mr Varadkar, a gay man, speak out against the traditional Catholic teaching on the family.