Newcastle tried to sign 19-year-old Barnes in January but their valuation was way below Derby County's.
United will hope the Rams' relegation will allow them to negotiate a more realistic price for the attacking midfielder.
Even when they turned down United's offer, believed to be around £1.5m, Derby are looking for money to rebuild.
Derby hoped Barnes would justify a reputation earned in the Football League, but he has failed to do so.
In a season restricted to just 14 league starts by form and fitness, Barnes has not stood out, scoring just once in the league and once in the FA Cup, against Championship side Sheffield Wednesday.