However, the Blues are expected to turn down the initial offer as they seek a larger loan fee.
Cucurella is open to a move to Old Trafford but negotiations are delicate with other left-back candidates having been sounded out.
United boss Erik ten Hag is thought to favour Cucurella, whom he believes fits his style of play.
The Spain international has been an unused substitute in all three of Mauricio Pochettino's first Premier League matches in charge at Stamford Bridge.
However, Cucurella is thought to be in contention to play for Chelsea against fourth-tier AFC Wimbledon in the Carabao Cup second round on Wednesday night.
When asked about the player’s future on Tuesday, Pochettino said: “I cannot talk about supposition or rumours. At the moment nothing changed, for tomorrow he is going to be able to play. The club did not communicate to me anything about this situation."
United are considering alternatives should they fail to reach a deal with Chelsea for Cucurella, including Tottenham's Sergio Reguilon and Lyon's Nicolas Tagliafico.