Jose Mourinho and Shaw’s relationship has been rocky since the manager’s arrival last year.
And the United boss has been looking at potentially bringing in a new left-back during the summer.
Tottenham star Danny Rose opened the door to a potential move by revealing he would be keen to move back up north.
However, Spurs still do not want to sell and Starsport understands United have abandoned their pursuit.
In the wake of this, The Times claim the Red Devils are expected to extend Shaw’s contract.
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“Mourinho believes he will be fit for selection next month”
The 22-year-old is now nearing a return to first-team action, with the defender back in full training at Carrington.
Mourinho believes he will be fit for selection next month, although no exact comeback date has been set.
Despite Manchester United having been keen on Rose and unsure over Shaw, Mourinho was impressed by him in pre-season.
Shaw travelled with the squad for the tour of America and caught the manager’s eye with his attitude.
The ex-Southampton ace has 10 months left on his United deal, meaning he could currently leave for free at the end of the campaign.
However, United can trigger a 12-month extension and are expected to do so.
The Times add United are unlikely to sign Shaw’s England international team-mate Rose in the coming fortnight.
And getting Shaw back fit is essential with Marcos Rojo sidelined until at least December.
Daley Blind and Matteo Darmian are Mourinho’s two makeshift left-back alternatives at the moment.