The Axios report cited anonymous sources, saying the proposed wave of strikes would be likely to include infrastructure targets.
Another plan focused on taking over part of the Strait of Hormuz so that it can be reopened for commercial shipping, Axios reported, adding that doing so could involve troops on the ground.
US-traded West Texas Intermediate crude oil also rose, up by 2.3% at around $109 a barrel.
The current Brent futures contract for June delivery is due to expire on Thursday. The more active July contract was up by about 2% at around $113 in morning trade in Asia.
Futures contracts are agreements to buy or sell an asset at a set date.