The private meeting is expected to take place around 0900 GMT, according to his diocese, with the pontiff to decide within weeks whether to accept his resignation.
The pope has previously defended the cardinal, saying in 2016 that his resignation before a trial would be “an error, imprudent”.
Barbarin is not expected to speak publicly.
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Fran cis said in February that “no abuse must ever be covered up, as has happened in the past”, or be under-estimated.
Barbarin, 68, is the most senior French cleric caught up in the global paedophilia scandal that has rocked the Catholic Church. He said after his
March 7 conviction that he would travel to Rome to tender his resignation.
The court in the southeastern French city of Lyon found Barbarin guilty of failing to report allegations that a priest, Bernard Preynat, had abused boy scouts in the Lyon area in the 1980s and 1990s.
The priest, who was charged in 2016, is expected to be tried this year.
Barbarin's lawyer immediately announced plans to fight the landmark ruling, which was hailed by abuse victims as ushering in a new period of accountability in the French Church. It remains to be seen however if the pope will accept his resignation ahead of the appeal judgement.
His trial comes as Francis battles to restore faith in the Church following a slew of abuse scandals that have spanned the globe, from Australia to Chile and the United States.
Less than a week after Barbarin's conviction the Vatican's former number three, Australian Cardinal George Pell, was sentenced to six years in prison by a Melbourne court for the “brazen” sexual abuse of two choirboys.
Barbarin, an arch-conservative who took over as archbishop in Lyon in 2002, was an outspoken opponent of gay marriage.
He had long been accused by victims' groups in Lyon of turning a blind eye to child abuse in his diocese, which blighted dozens of lives.
“I cannot see what I am guilty of,” Barbarin told the court at the start of the trial in January. “I never tried to hide, let alone cover up, these horrible facts.”
But the court found otherwise, saying the archbishop, “in all conscience”, chose not to tell authorities of the abuse allegations “in order to preserve the institution to which he belongs”.
Two other senior French religious figures have been convicted of failing to report child abuse in the past: Pierre Rican, the archbishop of Bayeux-Lisieux, in 2001, and the former bishop of Orleans, Andre Fort, last year.