The remarks from the US president came after the two sides on Sunday held their fourth round of talks that began last month and represented the highest-level contact since Trump in 2018 pulled out of a three-year-old deal.
Speaking in Qatar as part of his multi-day tour of the Gulf, Trump voiced optimism at avoiding a military strike on Tehran's nuclear sites.
"We're not going to be making any nuclear dust in Iran," he said. "I think we're getting close to maybe doing a deal without having to do this."
Both main crude contracts sank more than three percent.
The commodity had already been falling Thursday on signs Iran could agree to certain demands.
An adviser to supreme leader Ayatollah Ali Khamenei said Wednesday that Tehran could accept far-reaching curbs on its atomic programme in exchange for sanctions relief, according to NBC News.
In an interview with NBC News, Ali Shamkhani said Iran could agree to never develop nuclear weapons, give up stockpiles of highly enriched uranium and allow inspectors to nuclear sites -- among other steps -- if economic sanctions were lifted.
Shamkhani said "yes" in response to a reporter's question on whether his country would be willing to sign an agreement with Washington if sanctions were lifted "immediately".
Meanwhile, equity markets stuttered as investors await fresh developments in trade talks, with US partners looking to reach deals to avoid Donald Trump's tariff blitz.
With excitement from the China-US detente running out of legs, the search is on for fresh catalysts to drive a rally that has pushed markets back above the levels seen before Trump's April 2 "Liberation Day" bombshell.
News that Beijing was suspending some non-tariff countermeasures on US entities for 90 days following the superpowers' weekend truce did little to inject much more enthusiasm.
With the tariffs crisis calmed for now, dealers can turn their attention to hard economic data, hoping for an idea about the initial impact of Washington's trade policies.
After figures Tuesday showing US inflation came in a little below forecasts in April, eyes are on wholesale prices and retail sales due later Thursday, as well as earnings from retail giant Walmart.
However, analysts pointed out that the real impact would not be seen until May's figures are released and warned that there were still plenty of bumps in the road ahead.
"The trade truce may hold for now, but the tariffs announced -- many still around 30 percent -- are not disappearing," said Charu Chanana, chief investment strategist at Saxo.
"These are 'sticky' policies that can reshape supply chains, corporate margins, and even inflation. In fact, the market is now preparing for a second shock: weaker economic and earnings data in the third quarter as tariffs bite."
She added that "the muted market reaction the day after the truce suggests investors may be digesting the idea that 'the best news may already be out'".
Shares in Tokyo, Hong Kong, Shanghai and Seoul were all down, but Sydney, Singapore, and Jakarta rose.
London retreated as energy firms including BP and Shell tumbled on signs of progress in the Iran nuclear programme.
Traders brushed off data showing Britain's economy grew more than expected in the first quarter. The reading covered only the period before the announcement of Trump's 10 percent levies on Britain and finance minister Rachel Reeves' business tax hike.
Paris and Frankfurt also fell.
- Key figures at around 0810 GMT -
West Texas Intermediate: DOWN 3.3 percent at $61.09 per barrel
Brent North Sea Crude: DOWN 3.1 percent at $64.05 per barrel
Tokyo - Nikkei 225: DOWN 1.0 percent at 37,755.51 (close)
Hong Kong - Hang Seng Index: DOWN 0.8 percent at 23,453.16 (close)
Shanghai - Composite: DOWN 0.7 percent at 3,380.82 (close)
London - FTSE 100: DOWN 0.5 percent at 8,541.97
Euro/dollar: UP at $1.1195 from $1.1178 on Wednesday
Pound/dollar: UP at $1.3274 from $1.3268
Dollar/yen: DOWN at 145.87 yen from 146.65 yen
Euro/pound: UP at 84.33 pence from 84.21 pence
New York - Dow: DOWN 0.2 percent at 42,051.06 (close)
dan/fox
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