The 20-year-old is largely unknown in English football after moving to the United States before the start of his career, but the player has just been handed a call-up to the England Under-21 squad.
The youngster has netted 14 goals in 53 MLS appearances for NYCFC, and it appears that has alerted both City and Manchester United.
However, according to The Sun, City are able to sign Harrison ahead of United due to having first refusal given their association with the Stateside team.
Harrison is unlikely to link up with the City first team in the foreseeable future, but head coach Pep Guardiola has previously stated a desire to sign more English players and Harrison could fit his criteria given his age.
An international debut could be handed to Harrison in one of England Under-21s' upcoming matches with Scotland and Andorra.