The Nottingham Forest wideman has fallen out of favour at Forest under manager Aitior Karanka and made just four appearances in 2018.
Olympiacos are desperate to reshape their squad after they lost their Greek title, after they had previously had won seven titles in a row.
And Forest owner Evangelos Marinakis, who also owns Olympiacos, is in Greece this week to rubberstamp the deal, according to reports in his homeland.
Scotland international McKay, who was signed by Forest by former Rangers manager Mark Warburton in a £500,000 deal, is expected to agree terms this week too.
McKay made over 100 appearances for Rangers before moving down south. Now he could be heading abroad with the Piraeus outfit.