Ambrose joined Birmingham from Crystal Palace in the summer of 2012, but has made just 10 appearances during his time at St Andrew's.
He has featured only twice this season, but has not played since September and is out of contract at the end of the campaign.
Blues have confirmed that the 29-year-old has travelled to Greece to undergo a medical with Apollon, with a view to finalising a loan move.
Meanwhile, Birmingham boss Lee Clark is hoping to bring in new signings this month after loan trio Dan Burn, Jesse Lingard and Aaron McLean returned to their parent clubs.
Speaking after Saturday's 1-0 defeat at Brighton, Clark said: "The young players have been brilliant for me. I hope to add one or two to the group as well as having players come back from injury.
"We've lost Dan Burn, Jesse Lingard and Aaron McLean to their parent clubs and that's been a blow to us."