Brent crude futures rose 38 cents, or 0.6pc, to $64.61 a barrel by 1:04 p.m. EDT (1704 GMT). US West Texas Intermediate (WTI) crude futures rose 45 cents, or 0.8pc, to $57.96 a barrel, a 0.8 percent gain.
Iran on Monday threatened to restart deactivated centrifuges and step up its enrichment of uranium to 20pc in a move that further threatens the 2015 nuclear agreement that Washington abandoned last year.
Washington has imposed sanctions that eliminate benefits Iran was meant to receive in return for agreeing to curbs on its nuclear program under the 2015, deal with world powers.
The confrontation has brought the United States and Iran close to conflict, with US President Donald Trump calling off air strikes last month minutes before impact.
On Sunday, Trump issued another warning over Iran's nuclear activities. "They'd better be careful," he said.
Iran's Oil Minister Bijan Zanganeh said on Sunday that he was very hopeful of an improvement in the country's crude exports, state TV reported.
"We see enough possibility of military conflict to cushion renewed price declines that might be driven by mounting expectations for a major slowing in the global economic growth path," Jim Ritterbusch of Ritterbusch and Associates said in a note.
Oil prices remain under pressure from lingering worries about demand as the US-China trade war has dampened prospects for global economic growth.
Japan's core machinery orders fell for the first time in four months in May, the biggest monthly drop in eight months in a worrying sign that global trade tensions are taking a toll on corporate investment.
Goldman Sachs said growth in US shale production is likely to outpace that of global demand at least through 2020 and limit gains in oil prices despite output curbs led by the Organization of the Petroleum Exporting Countries.
In a sign of growing concern over demand, investors cut their ICE Brent net long positions by 6,748 contracts to 248,006 in the week to July 2.