The Gunners had resisted City's overtures for Alexis Sanchez all summer but finally relented and opened talks on deadline day, though demanded Sterling in return.
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However, the Gunners' £92m bid for the French global fell through and Sanchez was forced to stay at the Emirates, much to his disappointment.
City would like to land Sanchez in the New Year transfer window, but if Arsenal won't do business they will happily take him for free in the summer. Sterling has scored in City's last three Premier League games, finding the net against Everton, Bournemouth and Watford.
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The Guardian understood that a deal worth £55m (€62.3m) plus £5m in add-ons had been agreed between the Gunners and City for Sanchez to move to the Etihad Stadium, before the north Londoners pulled out a move for Lemar as the deadline approached. And with youngsters Marcus Rashford and Anthony Martial out on the wings, United's forward line has guile, pace and strength.
Bookmakers have reacted to news that United could be keen by slashing their odds from 10/1 into 9/2.
The ex-Everton man is believed to be happy at Old Trafford but with reported interest from China and Turkey, Mourinho fears his star midfielder could be persuaded to leave England for one last payday.
Though Guardiola described the prospect of a swap deal as "zero chance" earlier this month, the Citizens may be forced to consider the offer as it would pave the way for Sanchez's departure amid interest from both Chelsea and Manchester United in the ex-Barcelona and Udinese hitman. I did feel he became distracted.