Jose Mourinho has been monitoring the 18-year-old midfielder, who is seen by the Italians as one of their most prized assets, with a view to making a move for the player either in January or at the end of the season.
However, Koeman thinks he can steal a march on his Premier League rival with the offer of first-team football, while it is likely he would have to bide his time under Mourinho at United.
The Cypriot is yet to make a competitive first-team appearance for Juventus but the Serie A champions are eager to tie him down to a long-term deal.
However, having been one of the biggest spenders in the previous transfer window – most of it offset by the sale of Paul Pogba – Juve are thought to be willing to listen to offers for £17m and above, according to reports in Italy.
Kastanos joined Juve’s academy in 2014 and came to the attention of United’s rivals Manchester City when he scored twice against them in their Uefa Youth League clash last November.
Despite never making an appearance for his club, Kastanos has two caps for Cyprus, coming in either an attacking midfield position or drifting in from the left.