The meeting comes amid tension over Russia's military build-up in Syria and its annexation of Crimea last year.
The two leaders will meet on the sidelines of the UN General Assembly, a US official told the AFP news agency.
They are last reported to have spoken by phone after the Iran nuclear deal.
"Given the situations in Ukraine and Syria, despite our profound differences with Moscow, the president believes that it would be irresponsible not to test whether we can make progress through high-level engagement with the Russians," a senior administration official told the BBC.
The official said that the main purpose of the meeting from the US point of view was to ensure that Russia abides by the terms of the ceasefire in Ukraine negotiated in Belarus in February.
It is not clear if the meeting will take place before or after President Putin speaks at the UN, correspondents say. The Russian leader is also due to meet Japanese Prime Minister Shinzo Abe on Monday.
Relations between the US and Russia nosedived after the Russian annexation of Crimea in March 2014. The move led to the US and other Western countries imposing sanctions on Russia amid allegations that it was fanning the insurgency in east Ukraine by providing troops and arms.
Russia has strongly criticised the sanctions, describing them as a naked attempt to force President Putin from power.
Russia has also been accused by the US of becoming increasingly involved in the Syria conflict by sending weapons, troops and supplies to an airport near the coastal city of Latakia.
US Secretary of State John Kerry said on Tuesday that the build-up appeared to be limited to Russia protecting its own forces in the country.
His comments followed the publication of satellite images that suggested Russia was developing two additional military bases at a weapons depot and military complex north of Latakia.
US President Barack Obama and his Russian counterpart Vladimir Putin will hold their first face-to-face meeting in nearly a year on Monday in New York, officials from both countries say.