The 19-year-old may feature for Louis van Gaal's side against Newcastle on Boxing Day.
The £30 million summer signing from Southampton was ruled out for several weeks with an ankle injury at the end of November and it was anticipated that the United defender would not be fit until 2015.
Shaw could return ahead of schedule, however, which would mean a first-team appearance against Newcastle on Boxing Day.
Yesterday, the 19-year-old tweeted his desire to work as hard as possible to secure a speedy return to fitness.
And, according to the Daily Mail, the Manchester United man is on the cusp of returning to full training.
Should he prove his fitness at Carrington in the coming days, Shaw could feature for Louis van Gaal's side against Newcastle, their report states.
Before that, of course, Man United travel to Aston Villa tomorrow afternoon, while Newcastle host Sunderland in the Tyne-Wear derby on Sunday afternoon.