This is according to reports from The Daily Mail, which claim that Shaw - the £30 million-rated 18-year-old who has long been attracting attention from both of those aforementioned clubs - is seeking such talks with the Saints' hierarchy after the final game of the 2013/14 Premier League season and wants to find out just whether or not he will be allowed to depart St. Mary's.
It has certainly been a big campaign for Shaw, with the player having cemented his status as arguably the finest young prospect in the top-flight as well as positioning himself as a serious threat to established left-back duo Ashley Cole and Leighton Baines at international level.
With the World Cup in Brazil on the horizon, Shaw - a proud product of Southampton's relentlessly prolific and celebrated youth academy - has been an ever-present on the South Coast and his form one of the key reasons behind the club's success under Mauricio Pochettino.
A self-confessed Chelsea supporter, which club Shaw would be likely to join if indeed he were to depart Southampton in the near future is not certain.
While his allegiance to Chelsea is clear, both the Blues and Manchester United could have a need to invest in a new long-term option on the left-hand side of defence this summer, with the futures of Cole and Patrice Evra still yet to be fully resolved.
With Cole out of contract this summer, the England stalwart has often found it difficult to break into Chelsea's first XI in recent months.
Instead, Jose Mourinho - who brought Cole to Stamford Bridge from Arsenal during his first spell in charge - has often preferred to deploy reliable right-back Cesar Azpilicueta in his place.
Evra, meanwhile, is still a regular starter for managerless Manchester United, but looks set to depart Old Trafford this summer with his deal also running out.
Shaw would be a superb capture for both clubs, of course, but his future appears to rest on Southampton's imminent decision.
The Saints' final game of the campaign comes this Saturday, when they will face Manchester United at home.
Do YOU want to write for GiveMeSport? Get started today by signing-up and submitting an article HERE: https://www.givemesport.com/writeforgms