After being unconvinced by Alberto Moreno, Liverpool boss Jurgen Klopp decided to play James Milner as his makeshift left-back.
Milner did a capable job for the Reds last season but Klopp made it one of his aims to sign a recognised left-back and bought Robertson from Hull for £10m.
Shaw had a difficult 2016/17 campaign for United marred by injury, poor form and public criticism by his manager Jose Mourinho.
In truth, the England defender's United career has struggled to get going as a result of the double leg fracture suffered in 2015.
And according to The Sun, Liverpool looked to take advantage of Shaw's situation.
The report says Klopp had Shaw high on his list of transfer targets but his approach was knocked back with United reluctant to sell to their rivals.
Shaw has not featured for United in pre-season due to an injury suffered late on last season.
GETTY
Luke Shaw was wanted by Liverpool
But Mourinho is said to be impressed with the 22-year-old's efforts in rehab and his determination to be a United player.
However, The Sun still say Shaw fears he won't be a regular for United under Mourinho.
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