Ban Ki-moon asked the Argentine pontiff to "grace us with your presence at the United Nations headquarters at the earliest convenience", saying it would give him "an opportunity to speak of your vision".
The invitation came during a meeting between the 77-year-old pope and the secretary general at the Vatican in which Ban Ki-moon flagged up concerns such as the crises in Syria and Ukraine, growing intolerance between faith groups, and climate change.
"I see the need for calm, compassion, cooperation and courage. The papacy embodies these principles," he said, lauding Francis's "personal commitment to eradicate poverty and promote sustainable development."
The pope took the opportunity to raise the issue of abortion, after representatives of a UN committee this week said the Catholic Church's opposition may constitute a violation of the UN charter against torture.
In a strongly-worded speech, Francis called on the UN to resist the "economy of exclusion, the throwaway culture and the culture of death which nowadays sadly risk becoming passively accepted."
He urged it to have greater "awareness of the dignity of each of our brothers and sisters whose life is sacred and inviolable from conception to natural death."