Not a chance. Let's go from another angle. If the Mediterranean Ocean was blocked at Gilbraltar during the Ice Age it would have likely dried out quite a bit and have had a greatly reduced water level for perhaps thousands of years.(Many communities/cities would have been built on the shore line). If and when the water broke through the 'plug' at Gilbralter (there is scouring on the ocean flood to support this idea) it would have begun a slow flooding of the coastal communities of the Mediterranean ocean.
There would have been thousands (maybe millions) of refugees streaming all over the place escaping their flooded coastal cities.
Word would have reached Noah that the whole world was flooding. At some point, the plug at the Bosporus would have blown and began to create even more panic and refugees and certainly confirmed the reports and rumors of a catastrophic flood.
Noah, in his wisdom, could see that the Black Sea was rising at one foot per day and probably estimated the height at which he had to begin to build the Ark (big, big,ship) so that he could complete it before the rising water reached the building site.....and that's why it's way up on the mountain. The water never reached it because the Black Sea and the Mediterranean reached equilibrium with the waters of the worlds oceans.
Another possibility is that Noah could have started building the Ark when reports of flooding in the Mediterranean began to arrive. The flooding of the Mediterranean could/would have changed weather/climate patterns and created the 'forty days and forty nights' of rain story recorded in the bible and an explanation for all the flooding that was apparent.