According to Football Insider, the Clarets have moved one step closer to signing the defender.
The U’s player has attracted interest from multiple EFL clubs, but it now appears that Vincent Kompany’s side have moved into pole position to sign the promising centre back.
It has been an impressive 18 months for McNally since earning his breakthrough into the Oxford lineup.
The Irishman made 30 appearances in League One last season as the club finished 8th in the third division table.
It was an impressive season for the 22-year old, who even contributed with a haul of four goals to the team.
The first of which was his first league goal at senior level, scoring in a 3-1 win against Fleetwood Town in November of last year.
McNally has two years remaining on his deal with the club, but Oxford have agreed to a fee to part ways with the defender.
Quiz: Can you name which club these 27 ex-Burnley players are playing for now?
Despite being capped at underage level, he has also yet to make his debut for the Ireland national side.
A move to Burnley would be a big step up, with Kompany hoping to return the club straight back into the Premier League following their relegation to the Championship last season.
This would be an exciting next step for McNally in his career and will be a big chance for him to develop his game even further.
Burnley are in need of defenders, with Ben Mee and James Tarkowski both having left Turf Moor.
With Nathan Collins another possible player set to leave, there is space opening for McNally to potentially earn some significant playing minutes this season.
The defender is ready to step up to the second division, and should be eagerly anticipating the chance to prove himself at any even higher level.