It's understood Oar will fly out from Brisbane on Monday night for a medical with APOEL, who have triggered Oar's €350,000 ($515,000 AUD) release clause and need only to reach personal terms to secure the 25-year-old for next season.
Cypriot outlet Philenews says the club, coached by Dutchman Mario Been, is "very close" to striking a deal with the left-sided winger, who has another year left on his Brisbane contract but appears highly unlikely to see it out.
Oar spent five fruitful years in Europe playing for Eredivisie side FC Utrecht but returned to the A-League after an ill-fated stint with English Championship club Ipswich Town in 2015-16, citing homesickness.
He was set to become one of Brisbane's two marquee players for the coming season, along with Brett Holman.
But having failed to reach his peak in his second stint for the Roar the move could provide coach John Aloisi with a handy vacancy as he looks to rebuild his squad following the departures of attacking trio Jamie Maclaren, Thomas Broich and Brandon Borrello.
Oar, capped 34 times for Australia, holds a Spanish passport and would not be counted as a non-EU player for APOEL, who see him as an ideal replacement for Greek international Giannis Gianniotas.
Gianniotas was on loan from Olympiacos last season and helped APOEL reach the final 16 of the Europa League for the first time in the club's history.
APOEL are the reigning Cypriot champions, earning them a second qualifying round berth in next season's UEFA Champions League.