The Sun's Daniel Cutts reported the pair are both interested in the 20-year-old winger—who left United's academy for the USA aged 14—but the Sky Blues will have first refusal as both clubs are co-owned by the City Football Group.
According to Cutts, a source said:
"City will have first dibs on Harrison at New York if he continues his rapid progress.
"It's sure to rub United up the wrong way after what happened to him as a kid. He's having a great time over in the U.S.
"City are hearing positive things from the execs in New York and that Harrison is an exciting late bloomer."
Manager Pep Guardiola may even have the option of inviting Harrison to train at City in order to better assess his abilities.
The youngster has had an excellent season in MLS, contributing 10 goals and three assists mainly from the right wing, though he has been used sporadically on the left or down the middle.
His form has seen him called up to England's under-21 side, prompting his club to share his highlights so far this season:
Harrison is not only a goal threat, he's also a tricky dribbler, as OptaJack demonstrated:
The winger is evidently delighted at the prospect of his first taste of international football:
While there's still a fairly large gap between the standards of the Premier League and MLS, it would be intriguing to see how Harrison fares at a higher level given the performances he's put in.
However, it's unlikely he'd receive much of a chance in City's first team, given they already have the likes of Raheem Sterling, Leroy Sane and Bernardo Silva competing for places out wide.
Nevertheless, the Sky Blues could offer him a window back into English football and better prospects of further international call-ups in the future, perhaps even to the senior team should he continue to develop his game.