ROME—The last time the Catholic Church’s cardinals picked a pope, they wanted an outsider to overhaul a scandal-plagued Vatican. They got more of a disrupter in Pope Francis than many had bargained for.

After Francis’ stormy 12-year pontificate, some cardinals now say they want a successor who can steady the ship and defuse tensions between progressives and conservatives over divorce, same-sex relationships, priestly celibacy and other contentious issues.