The Daily Star Sunday says there is concern at The Etihad Stadium about the number of home-grown players there will be in their squad for the start of next season.
Premier League squads must include eight in a 25-man pool.
Southampton left-back Shaw, who made his England debut last week at the age of 18, is the most-wanted young player in the country.
City boss Manuel Pellegrini, preparing for today's FA Cup tie against Watford, is aware that he would face fierce competition for Shaw.
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