The Blues are believed to be mulling over a deal that could see them steal the left-back away from Old Trafford, according to the Mirror, despite United's desire to keep Shaw in the north west of England.
Chelsea's London rivals Arsenal have also been credited with interest in the England international and, if both are seriously intrigued with the possibility of signing Shaw, it could lead to all-out war between the three English giants.
Shaw had fallen out of favour at United under Jose Mourinho after the pair came to verbal blows earlier in the Portuguese boss' reign.
However, the ice has thawed in recent times and Shaw has enjoyed his fair share of first-team action in the first team - so much so that his performances have improved dramatically.
That has led to a change of heart by Mourinho over whether he should retain Shaw's services past his contract's expiry date this summer, and a new deal is now said to be in the offing for the 22-year-old.
Jose on @LukeShaw23: "He has been, for a few months, free of minor injuries that can stop [a player's] evolution and I'm really happy [with him]. I think he will be a Manchester United player for years." #MUFC pic.twitter.com/fXZAN7RzkJ— Manchester United (@ManUtd) 9 February 2018
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It is unclear if Shaw will be handed a contract that he deems worth enough of his talents - the full-back currently earns around £80,000-a-week - and Chelsea and Arsenal could use that hesitance on United's part to swoop for the former Southampton starlet.
Antonio Conte is, strangely, said to have concerns that first choice wing back Marcos Alonso could head back to his native Spain this summer and deprive him of an attacking defender in that role.
Arsenal and Chelsea got no chance of getting Luke Shaw. Take the L.— ManLikeAlexis (@JamesMUFC96) 14 February 2018
Arsene Wenger, meanwhile, is searching for a long-term replacement for veteran full-back Nacho Monreal despite bringing in Sead Kolasinac on a free from Schalke last July.
Monreal only has 12 months left on his own contract and, as 32 years of age, may want more job security elsewhere, while Kolasinac has failed to set the world alight since his move to England.
With United believed to be monitoring Fulham wonderkid Ryan Sessegnon, Shaw could find himself out in the cold again and a move back down south could be his best bet for regular senior football.