The United States is imposing new sanctions on Iran's oil industry after Washington withdrew from a nuclear deal between Tehran and other global powers.
Bolton said the administration understands that a number of countries, some close geographically to Iran which he visited last week, and others "may not be able to go all the way, all the way to zero immediately".
Separately, Oil Minister Bijan Zanganeh said on Wednesday 280,000 barrels of oil had been sold on Iran's energy bourse and an additional 720,000 barrels would be offered for sale again on the exchange, according to Oil Ministry news site SHANA.
The Trump administration on Friday announced the restoration of the final round of sanctions on Iran that were lifted under the controversial 2015 Iran nuclear deal - though there will be some exemptions for eight countries.
Iran began selling crude oil to private companies for export on Sunday as part of a strategy to counter the planned sanctions.
Countries seeking waivers - in a process similar to that under former President Barack Obama - are expected to agree to substantially reduce their Iranian oil imports.
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Iran's biggest oil customers - all in Asia - have been seeking sanctions waivers to allow them to continue buying some of its oil and have argued that a total ban would spur a further rally in the price of crude.
That's symbolically important to the Trump administration because President Barack Obama's administration took three years to remove 1.2 million barrels from the market - and that was while acting in concert with the European Union and other nations before the worldwide effort yielded the 2015 deal. "I think it's important that we not relax in the effort", he said. "There is much more that it's going to need to do", he said. "The people of Iran should know with certainty that the government has no fear of the new American threats".
On the other hand, it's unclear whether the United States would grant any waivers, with U.S. Administration officials giving mixed signals.
"And the history, since May, now nearly six months on, is that most European companies are choosing - not surprisingly - to do business with America".
Iran is India's third-largest oil supplier after Iraq and Saudi Arabia. Currently, Iran provides its tankers as well as insurance for oil cargoes to India.
In the first three weeks of October, Iran's crude and condensate exports averaged around 1.90-1.95 million bpd according to Platts ship tracking data, but exports could be even higher due to the Iranian tactic to switch off tracking devices to hide some volumes and destinations of its oil exports, sources say.