The 19-year-old has told Charlton Athletic that he wants to "explore his options elsewhere" leaving the Championship club "extremely disappointed".
Poyet, the son of Sunderland manager Gus, became a fan favourite at The Addicks last term after making 23 league and cup appearances from January onwards.
He was voted as the club's Player of the Year in May after his impressive displays helped Charlton keep their place in the Championship, and his decision to move on has left the South London club upset.
The Gunners are always looking to bring in the best youngsters and the central midfielder is certainly talented enough to make the move. However, there is little chance that he would see any playing time at The Emirates as a result of such fierce competition for places in the Arsenal midfield.
Wenger may still look to add the midfielder, who has represented England at youth levels, as a free transfer with the decision to loan him out next season the most likely outcome.
The Blues are another club who are always keen to add young talent but there are already a vast amount of talented youngsters at the club and a move for Poyet would not guarantee him any playing time in the near future.
His father Gus was a hero at Stamford Bridge though and he may want to follow in his footsteps.
Spurs are another of his father's former clubs, and The Lilywhites have some of the best young players in the country on the books.
Tottenham have placed plenty of emphasis on youth in recent years and will undoubtedly be considering a move for Poyet, though there are too many central midfielders already at the club, and Nabil Bentaleb seems to have nailed down any spot in the team that Poyet could have had.
The Premier League champions will be an attractive option for Poyet and they do not have as many options in the centre of midfield as the aforementioned clubs.
It is unlikely that Poyet will see much action with a move to The Citizens but with Yaya Toure's future at City far from certain, a move for the youngster might be perfect option on a free transfer.
Plenty of other clubs in the Premier League will be tracking the 19-year-old and a move to Sunderland to play under his father cannot be ruled out, though with some of the biggest clubs reportedly interested, it will be interesting to see where Poyet moves this summer.