The Red Devils have not put a timeframe on Shaw's absence, but it is believed that he could be missing for a couple of months, while the back-up left-back Tyrell Malacia is also currently on the sidelines.
Speaking during his press conference on Friday ahead of the Premier League clash with Nottingham Forest, Ten Hag said that Malacia would be absent for at least another "few weeks".
"Will also take a few weeks before he will be back. Definitely, we have the solutions in our squad to cope with that. We thought before the season about this scenario, it could happen, you hope to avoid. But if it happens you have to be ready for it," said the Dutchman.
According to the Manchester Evening News, Man United are currently looking at the left-back market, as Brandon Williams was allowed to leave for Ipswich Town on Thursday, further depleting the club's options in that area of the field.
Alvaro Fernandez is also an option, but the 20-year-old is widely expected to leave on loan, with La Liga outfit Granada said to lead the race for his services.
Diogo Dalot is expected to start at left-back against Forest on Saturday, but the club are believed to be looking to sign another player capable of operating in that area of the field.
According to Stretty News, Bertrand is being considered, and there has already been contact between the two parties over a potential short-term move to Old Trafford.
The report claims that there have been 'initial discussions' with Bertrand's representatives, and there is a chance that the 34-year-old could make the move to the Red Devils on a free transfer.
Bertrand is currently training with former club Chelsea, with his contract at Leicester City expiring at the end of June.
The 19-time England international started his professional career at Chelsea, making 57 first-team appearances for the Blues before moving on to Southampton in 2015.
Bertrand has a wealth of experience in the Premier League, making 262 appearances in the competition during his time at Southampton, Chelsea, Aston Villa and Leicester.
The defender did not make a single appearance for Leicester last season, though, as he was on the sidelines recovering from a knee operation that he underwent in the early stages of 2022.
Man United, meanwhile, remain in discussions over a potential deal for central defender Jonny Evans, who also saw his Leicester contract expire at the end of June.