Heading into the final few days of the transfer window, the Red Devils are searching far and wide for reinforcements in that area of the field due to Luke Shaw's recent injury.
The England international is facing several weeks on the sidelines due to a muscular injury, while backup option Tyrell Malacia is nursing a knee problem and will not be back until after the September international break.
With Brandon Williams joining Ipswich Town on loan, and Lisandro Martinez now settled in a central role, Diogo Dalot started on that side of the field in Saturday's 3-2 win over Nottingham Forest.
Erik ten Hag also has the young Alvaro Fernandez - whose chances of a loan exit have now been dashed - available to him, but the Dutchman is looking to add a new left-back to his squad before next Friday's deadline.
According to transfer expert Fabrizio Romano, Man United have sounded out Chelsea to discuss taking Cucurella on a temporary basis, although talks are at their earliest stages, and no formal offer has been made.
Romano adds that the Red Devils are not willing to break the bank on a new full-back, and Cucurella is one of three candidates on Ten Hag's shortlist, which includes two other unnamed defenders.
While the identity of Man United's other targets is unknown, ex-Chelsea and Southampton left-back Ryan Bertrand is allegedly under consideration, having departed Leicester City at the end of his contract in June.
However, the 34-year-old did not manage a single appearance for the Foxes last term due to a severe knee injury, and his only appearance came with the Under-21s in the Premier League 2.
Meanwhile, Cucurella turned out 33 times for Chelsea in his debut campaign last year, but he failed to recapture the sort of form that convinced the Blues to pay Brighton & Hove Albion £56m for his services.
The former Barcelona and Getafe man set up just two goals across all competitions, although he was deployed as a centre-back, left-back and left wing-back during the Blues' troubled campaign.
A muscular problem saw Cucurella miss Chelsea's final seven Premier League games last season, and he has been an unused substitute in each of the Blues' first three fixtures this time around.
Reports earlier this year claimed that a move to Atletico Madrid was a possibility for the 25-year-old, but his agent quickly rubbished those claims, while links with Tottenham Hotspur also led to nowhere.
Ben Chilwell has not been without his own injury problems, but the Englishman is above Cucurella in the pecking order when fully fit, while Ian Maatsen is also vying to make a name for himself on that side under Mauricio Pochettino this season.
However, Chelsea have already allowed Lewis Hall to join Newcastle United on loan with an obligation to buy and may be hesitant to let another left-back depart with less than a week to secure a replacement.