The International Energy Agency expects that the surplus will fall to 0.2m barrels per day (bpd) in the second half of this year, down from 1.5m bpd from January to June.
It comes as the decline in US output is gathering pace and Iran fails to add as many barrels as expected.
"The oil market looks set to move close to balance in the second half of this year. Oil prices are on the rise with Brent crude oil trading currently well above $40 per barrel," the IEA said in its monthly report.
Oil has collapsed since mid-2014 to as low as $27 per barrel, from as high as $115. This has been exacerbated by Organisation of Petroleum Exporting Countries' refusal to cut output to defend their market share, and drive the emergent US shale gas producers to bankruptcy.
Nevertheless, the IEA remained sceptical about the widely anticipated Opec and non-Opec oil freeze deal's ability to balance markets.
"If there is to be a production freeze, rather than a cut, the impact on physical oil supplies will be limited," it said.
"With Saudi Arabia and Russia already producing at or near record rates and very little upside seen apart from Iran any deal struck will not materially impact the global supply-demand balance during the first half of 2016."
Some of the the world's biggest oil producers will meet in Doha on Sunday to potentially firm up an earlier agreement between Qatar, Saudi Arabia, Russia and Venezuela which will result in a wider freeze.