The Daily Mail reports that Shaw has been placed on Brendan Rodgers's transfer shortlist as he plans for a Champions League campaign next term.
The Liverpool boss would have to compete with Chelsea and Manchester United for the 18-year-old defender, who won England honours for the first time earlier this month.
Chelsea and United have been the frontrunners since the January transfer window, when United reportedly had a bid turned down by the Saints.
The teenage full-back is just the latest star to be produced at the Southampton academy, and he looks poised to become their most lucrative yet with talk of a £30m valuation.
That may prove beyond the financial reach of Liverpool though, who may look to secure a cheaper deal if Southampton are willing to negotiate. However, that may be wishful thinking with so many top clubs interested in the player.
Since his debut against West Brom at the Hawthorns last season under Nigel Adkins, Shaw has gone on to become one of Southampton's most consistent players.
Moyes reportedly held a three-hour meeting with United chief executive Ed Woodward earlier this month to discuss summer transfer targets, and Shaw featured prominently.
All three of United, Liverpool and Chelsea need to sign a full-back at the end of the season, with Chelsea's Ashley Cole and United's Patrice Evra out of contract.
Liverpool meanwhile, have decided not to exercise their option to sign on-loan left-back Aly Cissokho.
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