The 30-year-old former Bournemouth defender has been used sparingly so far this season under Steve Bruce and he is thought to be considering his future.
The transfer window in Turkey doesn't shut until later today (FRI) and Goztepe are hopeful of getting a deal over the line, according to Turkish-Football.
Villa have already seen one player join a Turkish Super League club this summer, with Aly Cissokho signing for newly-promoted Yeni Malatyaspor.
And it would appear that Elphick may follow suit as the defender eyes a return to first-team action.