While markets for Brent crude trading do not open until 23:00 GMT on Sunday, so-called over-the-counter trades between private buyers suggest the price has jumped by around 10% to $80 (£59) a barrel.
Some analysts have warned it could go over $100 in the event of a prolonged conflict.
On Sunday, the Opec+ group of oil producing nations – which includes Saudi Arabia and Russia – agreed to increase their output by 206,000 barrels a day to help cushion any price rises, but some experts doubt this would help much.
Edmund King, president of the AA, warned the disruption could drive up petrol prices around the world.
“The turmoil and bombing across the Middle East will surely be a catalyst to disrupt oil distribution globally, which will inevitably lead to price hikes,” he said.
“The magnitude and duration of pump price increases depends on how long the conflict goes on.”