The club are lining up £40m bid for the England international and feel this should be sufficient to force Everton’s hand in spite of their reluctance to sell.
City scouts were there to watch Barkley’s return to the Everton team after his recover from injury, and the midfielder again impressed in the 3-0 win over Aston Villa.
Chelsea have also been following the 20-year-old’s progress, with both clubs having enquiries rebuffed over the summer.
Martinez denied that there had been any concrete offers for Barkley but said that the Toffees would take their interest as ‘a compliment’.
Barkley is particularly appealing to Manchester City as he would help them achieve their quota of homegrown players which was negatively affected with the summer departures of Jack Rodwell and Joleon Lescott.