Swift, who joined the Royals in 2018, has started in each of their opening eight games of the Championship season, registering seven goals and providing four assists to sit atop the league standings for both statistics.
Steve Bruce's side will face competition for his signature though, with Aston Villa and Sheffield United said to interested in the creative midfielder alongside Brentford, who could swoop to bring him back to the club upon their promotion to the Premier League.
The Magpies' only incoming transfer in the summer was Arsenal midfielder Joe Willock, and Bruce has spoken on the need to improve his playing squad.
The St. James' Park side currently occupy 18th place in the English top flight, having not won any of their five league games so far, while only collecting two points. body check tags ::