When was Homer's poetry most probably written down?

Yes, or at least the first known poet, because he was in Mesopotamian times.

Answer:

Although the Mesopotamian poem "Epic of Gilgamesh" was written about 2000 BCE no author is indicated. The earliest version we have was put together by Sin-liqe-unninni a Mesopotamian in the period between 1300 BC and 1000 BC. Homer's poem, or at least poems attributed to the Greek Homer were written in 850 BCE (maybe) but hey too may have been put together at a later date by others.

The earliest poem with recorded poet(ess) is also the oldest love poem. It was recited by an unnamed bride of the Sumerian king Shu-Sin, who ruled from 2037-2029 BCE.