The Sky Blues were linked with a £25m swoop for the 20-year-old over the weekend, amid rumours that Manuel Pellegrini is keen to draft in top English talent to fill his squad’s home-grown quota.
Barkley is widely seen as one of the nation’s top emerging talents on the back of his fine displays in the Premier League and impressive showings at the World Cup.
The youngster himself is understood to be happy at Goodison Park – where he is a regular starter – but there has been talk that a massive offer may turn the player’s and the club’s heads.
But now reports suggest that Everton will not alter their stance that the attacking midfielder is staying put.
City were thought to be willing to make Jack Rodwell a part of a potential deal, but the Toffees may opt to launch a separate move for their former star.
The 23-year-old swapped Merseyside for Manchester in 2012, but has been unable to cement himself as a first-team regular due, in part, to a series of injuries.
He is understood to be keen to head back to his old club, and came close to agreeing a loan deal in January.