The pontiff expects to meet with Russia later in the month because it is the current leader of the United Nations Security Council.
The role of leader is taken in turns between the five permament members, which also include Britain, France, the United States and China.
The meeting between Pope Francis and Vladimir Putin is scheduled two days before a United Nations development summit, where the pontiff will be speaking on the refugee crisis, the environment, and other world issues.
However the Russian Mission, which represents Russia, has said Mr Putin is unlikely to be able to arrive two days before the development summit, making their private meeting uncertain.
No information has been released on what the pair would talk about if they were to meet.
Pope Francis has two other bilateral meetings at the UN, both with the current and incoming presidents of the General Assembly, from Uganda and Denmark.
The last Pope to speak at a General Assembly was Pope Benedict XVI in 2008.
The Vatican's permanent delegate to the United Nations, Archbishop Bernadito Auza, spoke to the New York Times about Pope Francis' upcoming talks at the development summit: "There are so many big themes, and he has only 30 minutes to talk."
He added that the pontiff would stress the UN should "consider these people [refugees] as people like us."
The Pope recently pledged to take in two refugee families, and urged the rest of the Church and world to do the same where possible.
