A muscular problem is set to keep the England international on the sidelines in the coming weeks, while Tyrell Malacia is out until at least the end of September with a muscular issue of his own.
Man United also allowed Brandon Williams to leave on loan towards the end of last week, with the academy product making the move to Championship outfit Ipswich Town.
Diogo Dalot can cover the left-back spot, while Alvaro Fernandez is also an option, but the latter is expected to leave on loan before the end of the transfer window.
A recent report claimed that free agent Ryan Bertrand, who is currently training with former club Chelsea following his release from Leicester City, has emerged as an option.
Ten Hag, speaking during Friday's press conference ahead of the clash with Nottingham Forest, suggested that the club would bring in a left-back if the right opportunity cropped up.
"It can be, because we have to anticipate situations. So if there are opportunities, yes. But it has to be the right player," said the manager when asked whether Shaw's injury had impacted the club's plans.
"And otherwise we have to deal with the current squad. And, as I said before, I know we went through the scenarios before the season with our transfer strategy and if that happened, we have the solution."
Ten Hag named Dalot at left-back for Saturday's game against Forest, with Fernandez on the bench.
There have also been suggestions that Man United could move for a new forward before the end of the transfer window, but the club's head coach has appeared to rule out an attacker arriving.
The Red Devils could still potentially make two more signings before the end of the market, though, with a goalkeeper and central midfielder on the agenda.
Fiorentina's Sofyan Amrabat is believed to be the club's leading target in the middle, but it is understood that an offer has still not been submitted to his Italian outfit.
Meanwhile, Fenerbahce's Altay Bayindir is expected to arrive to provide competition for Andre Onana between the sticks, which is likely to lead to Dean Henderson moving on next week.