OPEC agreed on Friday to raise oil production by around 1 million barrels per day from July for the group and its allies, an OPEC source said.
A production increase on Friday would undo a cut agreed on in late 2016 that has since then helped push up the price of oil by nearly 50 percent.
Iran has so far been the main barrier to a deal, with Zanganeh saying on Wednesday OPEC should not yield to pressure from U.S. President Donald Trump to raise output.
Speaking at the same event, Iranian Oil Minister Bijan Namdar Zanganeh reiterated his country's resistance to hiking oil output and accused Trump of trying to politicise OPEC.
But it remains to be seen what kind of compromise could emerge, given Iran's deep aversion to any attempt by OPEC allies to offset its expected production losses as a result of the impending U.S. sanctions.
OPEC energy ministers scrambled to hammer out a compromise on oil output policy Friday as they debated a Saudi proposal to ramp up production that has encountered fierce resistance from sanctions-hit Iran.
According to Bloomberg, the warning came after Russian Energy Minister Alexander Novak said that "Moscow and Riyadh have agreed to cooperate in the oil market".
The prince, who is Saudi minister of state for energy affairs, was speaking at the OPEC seminar in Vienna, ahead of what is likely to be a hard meeting.
Other OPEC members, including Iran, oppose such a move, fearing a price slump.
Refinery crude runs rose to 17.7 million barrels per day, the highest on record for this time of year, the EIA data showed.
Patrick DeHaan, an analyst for GasBuddy, a gasoline price tracking service, said a middling OPEC increase - more than 600,000 barrels per day - would make it less likely that Americans will pay an average of $3. When it's too low, oil companies cut back operations and lay off thousands of workers.
Commercial U.S. crude inventories dropped by 5.9 million barrels in the week to June 15, to 426.53 million barrels, the EIA said.
"President Trump thinks that [he] can order to OPEC and instruct to OPEC to do something", he said.
A 25% tariff on U.S. crude oil imports, as threatened by China in retaliation for duties Washington has announced but not yet implemented against Chinese products, would make American crude uncompetitive in China versus other supplies.
Officials from major oil-producing nations are expected to agree this week to boost output, but just how much they will open the spigot - and the effect on oil prices - remain wild cards.
OPEC's decision confused some in the market as the producers gave opaque targets for the increase, making it hard to understand precisely how much more it will pump.