The 19-year-old Danish international has clubs across the continent alerted to his predatory talents after grabbing a first-team opportunity this season.
Arnesen has hit eight goals in 12 outings and Ajax are desperate to tie him down to a new long-term deal.
City, however, have now gone public with their interest, which won’t be received well in the Netherlands and can be seen as the laying down of an aggressive marker by the Manchester giants.
Arnesen, the son of former Chelsea director of football, Frank, has been a regular at the Amsterdam Arena in recent weeks.
He confirmed to Danish station, BT: “We are keeping a very close eye on him (Dolberg), and when the time is right we’ll make a decision on whether we should act.
“He has a huge potential. He is tall, strong, brave and quick. He has an excellent touch, can play with his back to goal and make the right decisions.
“And he can finish anywhere in the field. He is so special, and therefore he is of course also on the radar.”
City seem determined to add to their collection of young striker talent, having already signed South American prospects Gabriel Jesus and Marlos Moreno, in addition to current first-team starlet Kelechi Iheanacho.
Rangers centre-back Russell Martin will miss Scotland's friendlies against Costa Rica and Hungary through injury.