City completed a historic season by winning their first-ever Champions League title on Saturday night.
Rodri's goal sealed a 1-0 victory over Inter Milan as City matched Manchester United's treble-winning season of 1998/99.
As well as the FA Cup, Pep Guardiola's side also claimed the Premier League title after a season-long race with Arsenal.
The Gunners were top of the league for 248 days, a record without winning the title, before City overtook them in the final weeks.
Zinchenko joined Arsenal from City in the summer of 2022 after six years at the Etihad.
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He was part of the City team which lost to Chelsea in the 2021 Champions League final.
The Ukrainian posted a series of 'clapping' emojis on Instagram alongside an image of City's Champions League triumph.
Given Arsenal were among the victims of City's treble success, some supporters did not appreciate Zinchenko's apparent delight.
One fan wrote on Twitter: "Trying to ignore that Zinchenko post but that's unacceptable I'm sorry."
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Another said: "Nah sell this guy asap…"
However, a third supporter tweeted: "If anyone has shown they care about Arsenal in the last season it's Zinchenko. I don't mind him being supportive of his former teammates."
Zinchenko's Arsenal teammate Gabriel Jesus, who also joined from City in 2022, did not react to the Champions League triumph on social media.