Greater Manchester Police said the arrests were made following 'separate incidents' during the game.
No further details have been revealed.
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The ages of both boys - or which team they support - has also not been disclosed.
GMP also confirmed a third arrest in connection with the game.
A man in his 20s was held on suspicion of a breach of the peace and police assault.
"All three remain in custody for questioning," a force spokesman said.
They could now face possible football banning orders as well as criminal charges, the force added.
Match commander, Chief Inspector Jamie Collins, said: "The majority of fans attending this afternoons match behaved in an exemplary manner and we thank those people.
"We work closely with Manchester City Football Club to promote good behaviour at matches and to identify anyone who commits a criminal offence.
"I want to send a clear message to anyone attending football matches intent on engaging in abusive and threatening behaviour that it will not be tolerated.
"GMP officers will take positive action against those using homophobic or racist language and that is what my officers have done at today's fixture.
"As well as being arrested we will work with our football partners to ensure the strongest action is taken including banning those fans from attending future football matches.
"Our top priority is the safety and well-being of the fans, staff and players and we want fans to be able to enjoy matches without the experience being ruined by a small number of people."
No further details of the exact nature of the incidents that led to all three arrests have been revealed.
City won the game 2-0, with goals from Bernado Silva and Kevin De Bruyne.
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