The summit recalls the April 2002 meeting John Paul called with the senior United States church leadership after the sex abuse scandal first exploded publicly in Boston.
The Vatican is grappling with claims that the leader of the Catholic Church helped cover up an American cardinal's misconduct.
He said a decision about his future "is an essential aspect so that this archdiocesan church we all love can move forward".
Mr DiNardo has said recent accusations of top Vatican officials, including the current pope, covering up for McCarrick deserve answers.
House Speaker Paul Ryan (R-WI) commented Thursday on the sexual abuse and cover-up scandal that is rocking the Catholic church.More news: Next month, Verizon will launch the first commercial 5G home internet service
In an embarrassing coincidence, Bishop Bransfield's cousin, Father Brian Bransfield, was among those meeting the Pope.
As the moment the United States cardinals arrived for their meeting with Francis in the Vatican's apostolic palace, the Holy See announced the Pope had ordered Archbishop of Baltimore, William E. Lori, to investigate harassment allegations against West Virginia Bishop Michael Bransfield.
"My primary concern is for the care and support of the priests and people of the Diocese of Wheeling-Charleston at this hard time", Lori said in a statement.
Last month, the Vatican expressed "shame and sorrow" over the sexual abuse of at least 1,000 children by more than 300 Catholic priests in Pennsylvania.More news: U.S. threatens ban to fight teenage vaping epidemic
The USCCB headquarters in Washington, D.C., released a short statement more than three hours after the meeting saying the pope had "listened very deeply from the heart" as they spoke to him in "a lengthy, fruitful and good exchange" about how the USA church had been "lacerated by the evil of sexual abuse".
Those allegations against Bransfield are not included the statement.
The petition calls for a response to allegations from Archbishop Carlo Maria Viganò that the pope learned of sexual abuse accusations lodged against former Cardinal Theodore McCarrick in 2013 and did not take subsequent action.
The Pontiff will meet with top officials from the Catholic Church early next year, the Vatican press office confirmed this afternoon.
Bransfield denied the allegations, and the Wheeling-Charleston diocese issued a statement in July 2012 saying they had identified the boy in the auto and he had denied any abuse. That man also said Gana told him Bransfield was having sex with a boy they saw him driving with.More news: Apple launches the iPhone XS and iPhone XS Max
"I pray and hope that the church gets this right", Ryan emphasized of the situation overall, "I think Cardinal DiNardo who is the head of the USCCB...from what I can see is on the right track and I pray for him as he goes to the Vatican".