Cole was released by West Ham in the summer, but Allardyce is now ready to re-sign the striker to bolster his attacking options.
"We've had Carlton training with us for a while now and as soon as he's fit enough to play there will be an offer of a short-term contract on the table, there's no doubt about that," Allardyce said.
"That's why he's here with us and that's why we're trying to improve his fitness having missed pre-season.
"As soon as he's okay and suitable terms and conditions have been agreed, which is nothing do with me, then he'll join us for however long the contract is negotiated for.
"We'll play a couple of behind-closed-doors games just to let him get some match practice and then we'll take it from there."