Brent crude futures rose 15 cents, or 0.2 per cent, to $ 76.46 per barrel at 0146 GMT, from their last close.
US West Texas Intermediate (WTI) crude futures were at $ 70.70, up 19 cents, or 0.3 per cent.
The United States demanded all countries stop imports of Iranian oil from November, a State Department official said on Tuesday.
Oil markets did not react more strongly to Washington’s pressure as the move was expected.
In addition, top exporter Saudi Arabia plans to raise output to make up for lost supplies.
“It is very unlikely the US will succeed in ending Iranian oil sales on this timetable, but we are increasing our estimate of oil likely to come off the market by November to about 700,000 barrels per day (bpd) – another bullish factor for prices,” risk consultancy Eurasia Group said.
During the last round of sanctions, which ended in 2016, several Asian countries received waivers from Washington allowing them to continue to imports from Iran.
This time, Washington already hinted when announcing renewed sanctions in May it was unwilling to grant waivers, and buyers from Japan, South Korea and India have already started dialling back purchases.
Beyond looming sanctions, other supply disruptions are tightening oil markets.
In Libya, a power struggle between the official government and rebels has left it unclear who will handle the country’s large oil exports, although as of Tuesday the oil ports of Hariga and Zueitina in eastern Libya were working normally.
In North America, a supply outage at Syncrude in Canada has locked in 350,000 bpd of crude, with repairs expected to last at least through July.
Stephen Innes of futures brokerage OANDA said the outage had contributed to a major draw in US crude oil inventories.
The American Petroleum Institute (API) on Tuesday reported a 9.2 million barrel reduction in US crude inventories in the week to June 22 to 421.4 million barrels.
Trying to make up for disrupted supply, the Organization of the Petroleum Exporting Countries (OPEC) said late last week it would increase output.
Top exporter and de-facto OPEC leader Saudi Arabia plans to pump a record 11 million bpd in July, up from 10.8 million bpd in June.
Despite this, French bank BNP Paribas said the “agreement to elevate output still leaves production restraints in place, limiting the market’s ability to rebuild inventories.”
“Considering significant future supply losses faced by Iran (under US sanctions) and supply risks in Venezuela and Libya … oil fundamentals still remain favourable for oil prices to rise over the next 6 months despite the OPEC+ decision,” BNP said.