Goalkeeper Byrne, 14, made his first-team debut for Irish club Glenavon last month in the semi-final of the Mid Ulster Cup.
According to the Mirror, United are confident of sealing his signature.
That is despite competition from Premier League rivals Everton, Liverpool and Tottenham.
But the Scouts in Attendance Twitter account reports that Championship giants Leeds have taken Byrne ‘on trial’.
Glenavon chairman Adrian Teer said: “There has already been a lot of interest in Conner from a number of cross-channel clubs.
“He is a great prospect.”
Leeds have put an increased focus on strengthening their youth ranks following Andrea Radrizzani’s takeover.
The Whites snapped up 16-year-old striker Ryan Edmondson from York City last month.