The champions are interested in attacking midfielder Barkley, 20, who excelled last season.
Barkley proved his pedigree while with England's World Cup squad, and started the final match of their group against Costa Rica.
City are aware they need to build up the number of 'home grown' players after a summer clear-out that will see Gareth Barry, Joleon Lescott, Micah Richards, Scott Sinclair and, possibly, James Milner leave.
City's unsettled former Everton midfielder Rodwell, 23, is keen to return to Goodison Park after becoming frustrated by a lack of first team opportunities, and that is something Barkley would need to consider.
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