Both sides are known to be seeking replacement left-backs for their current ageing bastions, Ashley Cole and Patrice Evra, both coming to the ends of their contracts at Stamford Bridge and Old Trafford, respectively.
Both clubs might favour engineering a move for Shaw's services right now, but it appears we'll have to wait until the end of the current campaign for any transfer to come about.
That's after it was reported by The Mirror's John Cross that the 18-year-old is willing to move in the summer, but won't leave Mauricio Pochettino's side in a lurch this winter.
An excerpt from the report reads:
United have been pushing to get a deal done in this winter window and there is a small chance of the deal happening before Friday's deadline if either club matches Southampton's valuation.
However, Shaw is willing to stay put - safe in the knowledge that Jose Mourinho's Blues will come back in for him in the summer. Shaw, a boyhood Chelsea fan, has set his heart on London rather than Manchester when the time comes to move.
It's admirable resolve from the teenager, who couldn't be blamed for wanting to jump at such an opportunity given that he's only been playing Premier League football for 18 months.
Irrespective of that fact, Shaw hasn't allowed a lack of experience get the better of him, and has taken the top-flight challenge incredibly well, starting in 20 of his side's first 22 league games of the season, per Transfermarkt.
Rather than disrupt that and undoubtedly fall behind someone else in the pecking order of a Premier League big gun, the defender seems intent to see his development carry on with its current worth in terms of team prominence.
Wayne Veysey of Goal.com says that the youngster would be an ideal purchase for David Moyes, and adds that he should be on the flight to Brazil this summer:
However, both teams still have an obstacle in their way in that the summer offers no security over their hopes of buying the youngster.
Pochettino is reported by talkSPORT as saying: "Southampton has no interest in selling him - not at this moment or in the summer. We have no intention."
A move from either side now doesn't hold any other positive other than ensuring a rival can't steal in for his signature come the end of the season, with each side possessing a player at left-back who's very arguably still a better option, despite Cole and Evra's advanced years.
However, with both stalwarts looking plausible to leave their stomping grounds in the coming months, an intriguing free-for-all may await, although the Blues would be seen as holding a slight advantage in that Shaw is a known Chelsea fan.