The offer made to Chelsea, according to sources, includes a small loan fee and coverage of Cucurella's wages.
The 25-year-old is surplus to requirements at Stamford Bridge just a year after his £55 million ($69.5m) move from Brighton. Cucurella is yet to feature under Mauricio Pochettino this season and Chelsea have been open to letting him leave all summer.
United are in the market for a left-back after losing both Shaw and Malacia, and allowing third-choice Brandon Williams join Ipswich Town on loan.
Financial Fair Play restrictions mean any deal would have to be as low-cost as possible. Erik ten Hag has already spent more than £160m this summer to bring in André Onana, Mason Mount and Rasmus Højlund.
As well as left-backs, United are also looking at midfielders in the final days of the transfer window, although if a deal happens it's likely to be on loan.