A possibility, but as nwr points out, the literalists insist that the flood was a global flood, and so must have contained representatives of all the "kinds" of animals.
Your scenario also presents some problems as well. If there were a flood in the Middle East, there should be some pretty clear evidence of it in the archaeological record. After all, floods leave some pretty distinctive sediments behind. It should be possible to find a single layer of flood deposits throughout the Middle East that dates to a single time.
Also, is it possible to flood the Middle East for a year, as Genesis states? The Middle East is not a bowl which can hold water -- the rain would pretty much flow away without building up overmuch. You can get flash floods in the valleys, but these would be very local, very temporary events. In this case, there would be no need to preserve the animals (they would not be in danger of extinction), nor would a boat be necessary -- Noah could have been warned to walk out of the area.
The flood is an interesting, and perhaps educational, story as a fable, but no matter how you look at it it is pretty ridiculous as an actual historical event.
Sort of like how, on another thread, someone is trying to claim that the Genesis creation story somehow describes the actual origin of the universe and the earth as modern science understands it. At some point you just have to realize that there is zero historical correspondence between various events described and actual history, and then just accept the stories for the value that they have, if any, as moral fables.
"The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one." -- George Bernard Shaw