The oil price rallied after Israeli Prime Minister, Benjamin Netanyahu, accused Iran of lying "big time" about a secretive nuclear weapons programme, ratcheting up tensions in a region that controls nearly half the world's oil.
Trump has given Britain, France and Germany a May 12 deadline to fix what he views as the flaws of the 2015 nuclear deal, or he will reimpose sanctions.
Brent crude futures were last down 2 cents at $74.62 a barrel by 1422 GMT, up from a session low of $73.47.
"Netanyahu may be the flavour of the day, but the larger issue driving this is what will Trump do with Iran?"
Oil prices rose on Monday after Israeli Prime Minister Benjamin Netanyahu stepped up pressure on the United States to pull out of the 2015 nuclear deal with Iran, presenting what he called evidence of a secret Iranian nuclear weapons programme.
"We'll see oil fluctuating as uncertainties will persist over whether Trump will withdraw from the Iran nuclear agreement", Vincent Hwang, a commodities analyst at NH Investment & Securities, said by phone in Seoul.
West Texas Intermediate crude for June delivery climbed 47 cents to settle at $68.57 a barrel on the New York Mercantile Exchange. "We therefore see oil prices as being well supported".More news: The Royal Baby's Name Is Making Disney Fans Really Excited
Israel shared with the US documents on a secret Iranian nuclear-weapons program and the USA has verified their authenticity, according to a person familiar with the matter.
Buyers were pressing for lower levels in part because of good availability of crude in Europe and the US and a wide gap between Brent and the US WTI benchmarks. The contract fell 0.4 per cent last week.
USA drillers added five oil rigs in the week to April 27, bringing the total count to 825, the highest since March 2015, General Electric's Baker Hughes energy services firm said. At the same time, OPEC-led production cuts have continued to tighten global markets, despite record-setting USA crude output.
"There are worries that Iran's oil exports could fall by about 1 million barrels per day from current levels", said Tomomichi Akuta, senior economist at Mitsubishi UFJ Research and Consulting in Tokyo. US crude production rose by 46,000 BPD the previous week, to 10.59 MMBPD.
The 2018 budget proposal, submitted by President Muhammadu Buhari in November 2017, put the benchmark oil price at $45 per barrel, compared to $44.5 per barrel for the 2017 budget.
US drillers added five oil rigs in the week to April 27, bringing the total count to 825, the highest level since March 2015, General Electric's Baker Hughes energy services firm said.