The 21-year-old was apparently close to joining fellow Championship outfit Leeds United near the end of the January transfer window, but the move collapsed.
James was handed his Wales debut by Manchester United legend Ryan Giggs in November. His rapid dash for Swansea against Brentford in February's FA Cup fifth round captured nationwide attention, and he further supplemented his burgeoning reputation a month later with a 20-yard smash for his country against Slovakia.
James mostly wreaked havoc down the left wing for Swansea during the 2018-19 term, an area where Manchester United can already choose from the likes of Marcus Rashford, Anthony Martial, and Alexis Sanchez.
Manchester United appear to be seeking young British talent in the summer window. Percy and ESPN FC's Rob Dawson understand the club is ready to table a formal offer for Newcastle United's Sean Longstaff. The Geordie midfielder, 21, has made nine Premier League appearances since his Magpies debut in December.