Van Aanholt joined Sunderland on a four-year deal from Chelsea back in July and started every Premier League game for his new club until he dislocated his shoulder in the 3-1 win over Crystal Palace earlier this month.
It has since been announced that the left-back will be out for around two-and-a-half months, prompting Poyet into thinking about bringing in a replacement, should there be any complications with the Dutchman's recovery.
"I met [sporting director] Lee [Congerton] again to define a few things," he said.
"And let's say it [left-back] is a position that needs to be on standby.
"We need to have an option, just in case something doesn't work and Patrick's recovery has a setback.
"It all depends on how he recovers.
"If he doesn't have any setbacks, it's going to be the middle of January before he is training again.
"But he is a fit lad and, once the healing after the operation is finished, the speed of the recovery is up to him. We'll see.
"It's a big blow to have lost Patrick. There are not many left-footed left-backs in the world - and it is very difficult to find one that is available at this level.
"He was getting used to what we wanted because we were working him hard every day!
"So it's a step back. We have to cope with the situation."
