The 24-year-old, who has been capped five times by his country, is currently a free agent after his contract with Croatian side Rijeka expired in the summer.
Mavrias can play as either a right full-back, wing-back or winger and was on trial at Easter Road a few weeks ago.
Head coach Neil Lennon was suitably impressed by Mavrias' performances to offer him a contract and a deal is expected to be concluded by the early part of next week.
Hibs are keen to reinforce on the right-hand side of the pitch, with recent Australia recruit Martin Boyle set to miss the early part of next year on Asian Cup duty with his new national team. The Easter Road outfit also expect to lose striker Jamie Maclaren and midfielder Mark Milligan for all of January and potentially some of February.
Mavrias started his career with Panathinaikos in his homeland and made his debut in a Champions League clash with Rubin Kazan in 2010, becoming the youngest ever Greek to play in the competition.
He moved to Sunderland in the summer of 2013, then in the English Premier League, for Â£3million, but failed to make a tangible impact on Wearside, going back to Panathinaikos on loan as well as Fortuna Dusseldorf in Germany.
Mavrias - also known as Charis - left Sunderland two years ago and joined German club Karlsruher, where he also played alongside current Hibs striker Florian Kamberi, before moving to Rijeka last term.