Attorney Mitchell Garabedian, who represents 86 plaintiffs who accuse Law and others in the archdiocese in a civil suit of covering up abuse by defrocked priests and convicted paedophile John Geoghan, will question Law for three hours.
On Wednesday evening, the Boston archdiocese said it suspended another priest while it probes a 25-year-old allegation of sexual misconduct. On Tuesday, the LA archdiocese was hit by a class action lawsuit over sex abuse.
For Cardinal Law, the crisis has marked a sharp fall from his position as the pre-eminent US prelate.
Garabedian won permission to question Law after the archdiocese backed out of a $15m to $30m settlement in the Geoghan case in May.
The archdiocese reneged on the deal after its Financial Council ruled it was too expensive and would use up money needed to compensate other alleged victims.
Garabedian has been seeking to convince a judge the deal was binding and said his questions on Thursday would focus on that issue: "It's very important that we be able to show that agreements existed that can be enforced."
Geoghan is already serving a nine-to-10-year sentence for fondling a child and faces at least one more criminal trial stemming from allegations by about 130 people that he molested them during his more than 30 years as a Boston-area priest.
Geoghan's case sparked outrage among US. Catholics, but the scandal snowballed into a national crisis after documents released at his trial showed Law and others knew about Geoghan's history of sexual abuse but still moved him from parish to parish, where he continued his abuse.