The England international had a hit-and-miss season under Jose Mourinho last time out.
The Special One publicly criticised the left-back, despite the former Southampton man struggling with injuries.
And according to The Sun, Liverpool made a move for Shaw, 23, before landing Andy Robertson this summer.
However, United are refusing to sell Shaw, especially to the Reds.
The Sun add that the left-back fears he may never be a first-team regular under Mourinho at Old Trafford.
Ex-Hull defender Robertson joined Klopp's side last month and provides balance in the German's back four.
The Liverpool boss used James Milner at left-back last season after dropping Alberto Moreno.
Robertson was Klopp's third summer signing.
The defender followed Mohamed Salah and Dominic Solanke to Anfield.
