Oil prices edged up in early exchanges, a day after plunging more than five percent on optimism for a lasting agreement between the two countries after more than three months of conflict that rattled energy markets and revived inflation.
Tuesday's selling was ramped up by a report in The Wall Street Journal that Washington could ease sanctions on Iranian crude as part of the deal to end the war, allowing Tehran to immediately sell crude and refined oil products.
Attention now turns to Friday's official signing ceremony in Switzerland and the subsequent negotiations that will focus on the fate of Tehran's nuclear programme and a plan for the lifting of international economic sanctions.
US President Donald Trump has said the Strait of Hormuz -- through which a fifth of global crude usually passes -- would "completely open" once the peace agreement is signed.
Both main contracts have tumbled this month as hopes for a deal grew, though analysts warned that with so many hurdles ahead prices were likely to be sensitive to developments.
"The risks are skewed to the upside," said Fabien Yip, market analyst at IG. "Any failure at the 19 June signing to produce a durable and transparent agreement -- particularly on nuclear provisions -- could rapidly reverse the recent decline, as each prior false start has demonstrated.
"A sustained recovery in strait traffic remains the most credible evidence that the deal is holding."
Meanwhile, oil industry experts and shipping companies have cautioned that the restoration of normal operations after the strait's near shutdown will take time.
Equity markets fluctuated ahead of the Federal Reserve's first policy announcement under new, Trump-appointed boss Kevin Warsh.
While expectations are for the central bank to stand pat on interest rates, investors will be keeping a close eye on its post-meeting statement for an idea about the policy committee's thinking in light of surging inflation and a strong jobs market.
Data last week showed US consumer prices rose in May at their highest level for three years owing to the impact of surging oil costs caused by the war.
Some observers predict the Fed will eventually announce an increase before the end of the year, despite Trump's previous demands for cuts.
"Warsh... inherits the most divided committee in more than three decades, with three voting members already dissenting against the easing bias in April, while outgoing governor Stephen Miran again voted in favour of a rate cut," said Michael Krautzberger, chief investment officer for public markets at AllianzGI.
"The minutes suggested that the committee's centre of gravity has shifted in a more hawkish direction as uncertainty about the duration and economic implications of the Middle East conflict continues to mount."
He added that "resilient economic activity and indications that labour-market conditions are stabilising support a less accommodative policy outlook".
"In a rapidly evolving economic and geopolitical environment, we expect the (bank) to remain on hold this year," he said.
"But resilient growth, a stabilising labour market and increasing inflation pressures have shifted the balance of risks in a hawkish direction."
In early trade, Tokyo and Seoul rose again -- building on a tech-led rally -- while Shanghai, Sydney, Singapore and Jakarta were also up.
But Hong Kong, Wellington, Taipei and Manila edged down.
That came after a mixed day on Wall Street, where the Dow hit a fresh record high but the S&P 500 and Nasdaq retreated.
- Key figures around 0230 GMT -
West Texas Intermediate: UP 0.1 percent at $76.12 a barrel
Brent North Sea Crude: UP 0.1 percent at $79.01 a barrel
Tokyo - Nikkei 225: UP 0.8 percent at 69,926.08 (break)
Hong Kong - Hang Seng Index: DOWN 0.1 percent at 24,462.70
Shanghai - Composite: UP 0.1 percent at 4,093.78
Seoul - Kospi: UP 0.1 percent at 8,738.84
Euro/dollar: UP at $1.1613 from $1.1608 on Tuesday
Pound/dollar: DOWN at $1.3425 from $1.3427
Dollar/yen: DOWN at 160.34 yen from 160.45 yen
Euro/pound: UP at 86.50 pence from 86.45 pence
dan/mjw