According to James Nursey in the Mirror, Shaw has "won over" manager Jose Mourinho with strong recent performances and will be rewarded with an improved new deal. His current contract is worth £80,000 per week and runs to 2019.
It marks a stunning turnaround for the left-back, who was heavily linked with a move away from Old Trafford late last year.
Paul Hirst of The Times wrote in November that United were "ready to sell" the England international for £20 million in the January transfer window—around £7 million less than they paid for him in 2014—as Mourinho had "made it clear that he had no future at the club."
Just under a month later, the 22-year-old was handed his first start since April by the Portuguese in the Red Devils' 2-1 defeat of CSKA Moscow in the UEFA Champions League.
Shaw has since worked his way back into the first team with some fine performances, and he has started nine of United's last 13 matches in all competitions.
Per the Manchester Evening News' Samuel Luckhurst, Mourinho's attitude towards Shaw has changed significantly in recent months:
For the best part of 2017 it looked as though Shaw's time at Old Trafford would forever be seen as a disappointing wasted opportunity.
After signing from Southampton as an 18-year-old in 2014, Shaw made a fine impression and particularly stood out with some brilliant performances under Louis van Gaal at the beginning of the 2015-16 season.
But then he suffered a horrendous leg break against PSV Eindhoven in the Champions League that ruled him out for the rest of the season.
Last term he started only nine Premier League matches, and he could barely make the matchday squad in the opening months of 2017-18.
Finally, though, Shaw has been given a run in the side, and he looks set to be rewarded for his recent efforts.