Police cars, an ambulance, and a coroner were seen outside the building following an incident where a body needed to be examined, according to ERT officials.
Baldock had joined Panathinaikos in May after a seven-year spell with Sheffield United, relegated from the English Premier League to the Championship last season.
Of Greek origin through his father, he was called to the Greek national team in 2022.
He had 12 caps, but had not been selected for the team's Nations League match against England at Wembley on Thursday owing to injury, ERT said.
Baldock had featured for Panathinaikos in a 0-0 draw with Olympiacos in a Greek Super League match on Sunday where he played for 75 minutes.
'One of our own'
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Both the Greece team's official website and his club's social media sites on X, formerly Twitter, and Facebook, were blackened in tribute to the player.
Manchester United's England international defender Harry Maguire, a former Sheffield United player, posted "RIP" and a heartbreak emoji alongside an image of Baldock on Instagram.
Sheffield United changed their home page to a black and white photo of their former player with the words: "RIP George Baldock.
"Sheffield United Football Club is shocked and extremely saddened to learn of the passing of former player, George Baldock," said the club.
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Northampton, where Baldock had a loan spell in 2011, wrote on X: "We are deeply saddened to learn of the passing of former loanee George Baldock at the tragically young age of 31.
"George made five appearances in 2011 before playing for Oxford United and Sheffield United. Sending our sincerest condolences to George's family and friends."
Baldock's boyhood club MK Dons said they were "deeply devastated and saddened to learn about the shocking news".
"George started his career at Stadium MK, where he worked his way through the youth ranks and made his senior career debut at the Club before reaching the heights of the Premier League in later years.
"You will always be one of our own, George."
The Football Association in England wrote on X: "We are devastated to learn of the passing of George Baldock at the age of 31. Our thoughts and deepest condolences are with George's family, friends and team-mates at club and country."
Baldock was nicknamed "Furious George" by Sheffield United fans in tribute to his uncompromising style.
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"I just love winning. I hate losing and I even hate drawing," he told local media.
(This story has not been edited by staff and is auto-generated from a syndicated feed.)