The 30 year-old centre-back said farewell to his team-mates ahead of AEK's departure to Romania for the first leg of their Europa League play-off tie against SC Vaslui.
Liverpool have yet to formally agree a fee but it is understood that a figure of 3 million euros has been mooted Kyrgiakos's career has seen him play Eintracht Frankfurt and Glasgow Rangers.
"Never in my life have I hidden from the consequences of my choices," a statement from the 30-year-old said.
"I said last week that I would be an AEK Athens player for the next four years and I meant what I said at the time.
"Today, however, I had to make the most important decision in my career. Going back on my word was difficult, but this is life, and life can bring about strange things."
The deal would plug a hole in Benitez's injury-ridden defence. Stoke City's Ryan Shawcross had previously been a target for Benitez.
