United are exploring whether they can bring in a left-back before the end of the transfer window this week.
Shaw's injury is worse than originally feared and he is facing months on the sidelines while Tyrell Malacia has no timeline on his return.
It has left the Red Devils with no senior left-back, though Diogo Dalot did fill in against Nottingham Forest on Saturday.
And United are eyeing short-term deals for Cucurella of Chelsea and Tottenham's Reguilon.
Both are adept at playing left-back or left wing-back and are currently struggling at their respective clubs.
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Cucurella has not played a single minute this season so far and Mauricio Pochettino opted to bring on Ian Maatsen rather than the Spaniard after replacing Ben Chilwell against Luton Town on Friday.
And Reguilon is in a similar situation at Spurs having dropped down the pecking order behind Destiny Udogie, while he has not played for the club since April 2022.
United are only eyeing a short-term loan rather than any obligation to buy, which might put off Chelsea and Tottenham.
All three clubs are likely to be vying for a Champions League spot this season and neither London club will be keen to strengthen a rival and weaken their squads.
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Cucurella only signed for the Blues last summer from Brighton in a deal worth up to £62m.
However, the defender had a tough first year at Stamford Bridge as Chelsea sacked two managers and finished 12th in the Premier League.
Failing signing either Cucurella or Reguilon, another option that could sign a short-term deal is Ryan Bertrand after he was released by Leicester this summer, talkSPORT understands.
The 34-year-old is currently training at Chelsea, where he came up through the academy, to keep himself fit.
The defender did not play a single game for Leicester last season while he could unite with fellow former Foxes star Jonny Evans, who is set to pen a one-year contract at Old Trafford.
Ten Hag admitted that United could sign a Shaw replacement, saying: "It can be (that United sign a left-back).
"I think we always have to anticipate on situations. If there are opportunities then yes, but it has to be the right player otherwise we have to deal with the current squad.
"We went through the scenarios before the season and if it happened, we have the solution."