Futures gained 1.7 per cent for the week as the agency said that, despite OPEC's near-perfect implementation of the curbs it agreed to with Russian Federation and other allies, stockpiles edged higher because of supply increases the countries made before the deal took effect on January 1.
Saudi Arabia, OPEC's biggest producer, is said to favour extending the supply curbs when the group meets next month, in line with the views of fellow members such as Kuwait and Venezuela.
Demand growth for 2017 is now expected to be 1.3 million barrels per day, down from the IEA's previous forecast of 1.4 million.
"It can be argued confidently that the market is already very close to balance", the agency said in its monthly report.
Brent for June settlement climbed 3 cents to $US55.89 a barrel on the London-based ICE Futures Europe exchange.
The IEA said oil stocks in the Organisation for Economic Cooperation and Development industrialized countries fell by 17.2 million barrels in March, although inventories were still 300 million barrels above the five-year average.
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As with Wednesday's report from Opec in Vienna, the IEA sees the USA as the main contributor to net growth this year of almost 500,000 bpd after a dip last year of almost 800,000 bpd, with the U.S. expected to raise production by 680,000 bpd.
With 390,000 bpd lower output from non-Opec - including an unplanned outage in Canada and lower North Sea shipments - the total drop in world supply last month was 755,000 bpd, the IEA estimates.
Oil prices are still around 25 per cent higher than they were a year ago, and roughly stable over the past three months, despite frequent day-to-day volatility. "We have an interesting second half to come".
Still, independent analysts see cause for optimism.
Oil prices have stabilised in recent weeks and could be boosted further if the cartel chose to extend its output reductions beyond the summer, the Paris-based agency said.
Oil inventories fell in March but probably showed a rise in the first quarter of this year overall because oil consumers stockpiled crude before the OPEC-led cuts took effect properly, the IEA said. "If this is true, crude can not be far behind".