Israel is part of the Middle East. I dont know what definition of Asia some people are looking at, but Asia is everything from Afghanistan and eastwards. Thats what they teach in geography.
If the Middle East has to be classified in a continent it is included in Africa. Thats why we commonly say "Africa and Middle East" to describe the general region.
There is a broader definition of Asia which some use that includes the Middle East, but i think that approach is not as common.