[7][8] The officials of AEK decided to not complete the transfer of Georgian international goalkeeper Giorgi Loria and start the 2016–17 season with Giannis Anestis and Barkas.

[17] After returning from injury, Barkas' form was a highlight for AEK and he made some very impressive saves in the qualifiers against Celtic and MOL Vidi in the UEFA Champions League preliminary rounds.

[citation needed] On 10 August 2019, Montpellier made two official offers for the Greek international, but AEK rejected both of them, setting a price tag of €8,000,000.

[18] In the 2019–20 season, he registered 10 clean sheets in 20 games of the domestic championship, helping AEK recover from a poor start to the campaign.

[citation needed] Barkas signed for Scottish Premiership club Celtic in July 2020 on a four-year contract for a £5 million fee plus future incentives.

[22] During the 2021–22 season, Barkas appeared to become the No.1 under new boss Ange Postecoglou, but poor performances against FC Midtjylland and West Ham resulted in Bain taking the gloves for the Champions League.

Οn 4 June 2016, he was called by Michael Skibbe into the Greece national team after Orestis Karnezis was injured for the friendlies against Australia.