A drill down on the graphic in the link you posted shows Israel and Lebanon are on the African "Plate".
Along with folds within a plate, and faults within the folds. Israel, and Lebanon, ended up in a fold essentially in between two plates. Might as well throw in Jordan also getting screwed by divine providence or sumpthin. With Syria only marginally better.
So there is no inference to be made that Israel has onshore oil just because it's neighbors do.
There's this concept the whole ME is just floating on oil. Well, it doesn't. Then comes the "leap of faith" since the East Med may have oil, that means there must be oil onshore. No it means the geological history of the East Med doesn't match the history of the dry land. Transitive properties just don't work with geology. Just because there are large oil fields in the Gulf of Mexico, some offshore from Georgia, doesn't mean Atlanta is floating on oil, or by continuing transitive imagination, Nashville Tenn is also floating on oil. The geology is different.
The Jewel of the Mind is Colored with the Hue of what it Imagines