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The Argentinian was disputing a decision to give the Gunners a throw in during the first-half and when Massey-Ellis went to walk away, Aguero put his hand on her shoulder.
Fans took to social media to question why the forward was not punished for the incident, with one writing: "Rules are clear, it's a booking at least, or sending off."
"Aguero can’t put his Hands on an official like that," added a second.
"Since when was grabbing an official okay? Aguero very lucky to not get sanctioned", wrote a third.
"What is this from Aguero? Very strange thing to do, isn't it?"
"Surely Aguero can't get away with touching a lineswoman like that?!"
A rule brought in back in 2016 tightened the regulations to protect match officials and any physical contact should result in a yellow or red card, depending on how severe it is.
A statement on the Premier League website at the time says: "Premier League football, and the wider game in England, is known for being competitive and compelling, but also for its fairness and for being played in the right spirit.
"However, the English football authorities have noticed over the past four seasons that intolerable behaviour by players and managers in their competitions have reached unacceptable levels in certain areas.
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"The Professional Game Match Officials (PGMO) will take a stronger position and action towards unacceptable participant conduct.
"This will focus on behaviour towards match officials, with the aim of reducing disrespectful conduct such as aggressively challenging decisions or running from distance to confront an official."
Aguero was making his first appearance for City after the 32-year-old suffered a knee injury last season.