Carroll a doubt for new season
The 24-year-old played an integral role as Allardyce's men finished 10th in the club's first season back in the Premier League, but appears unlikely to be fit in time for the new campaign, which starts on August 17.
Allardyce, though, is refusing to panic and does not want to rush the England international back to full training ahead of schedule.
"Andy is still recovering. He has got four weeks or more to go before he starts full training," Allardyce said.
"His rehab is going pretty well at the moment and will be joining us in Germany, and hopefully he will be available, as will George McCartney, as quickly as possible.
"I hope it will be before (September) for both of them, but being fit and being ready to play are two different things - they may be fit from the injury, but they are unlikely to be capable of playing straight away.
"If we are going to put our lads through six weeks of preparation and then Andy gets fit with two days to go, you would not expect us to play him, would you?"
Allardyce has confirmed he wants to sign Estudiantes striker Duvan Zapata, but the move is being delayed by issues with the Colombian's work permit.
"It is whether we think we can get the work permit through, that is the problem," he added.
"The fact he is not a full international would mean we have to go to a hearing, and we have to decide if that is worth doing, which we will decide in the next 24 hours, but we would like to (get him) if we could."