Sunderland have been linked with West Ham United duo James Collins and James Tomkins.
Sunderland look set to have to battle against the drop this season, and they will be keen to add quality at the next available opportunity.
One of the departments most in need of strengthening is at centre-back, where Allardyce’s options look slim.
However, the 61-year-old boss insists the Black Cats can’t afford to narrow their search too much, and they should have an eye on improving every area of their squad.
“I don’t know if we can be position-specific for January,” he said, when asked whether he intended to sign a centre-back in the new year.
“Obviously, you target positions which are more important than others.
“But if somebody popped up in another position that wasn’t as big a priority, but could be a massive bonus to the club and could make us better, then you’re still going to try to sign him.”
Sunderland have been linked with moves for West Ham United duo James Collins and James Tomkins by the Daily Mirror.
The Hammers are said to be willing to relinquish Collins, but they would require a £10 million fee to sell Tomkins.
January could be vital for Sunderland, as Allardyce will be keen to bolster his options to ensure that they can get out of trouble.
The Black Cats have improved under Allardyce’s stewardship so far, but they are still in the relegation zone as things stand.