Berkovic was expected to return to the City squad for tomorrow's trip to Chelsea but suffered a slight injury on Monday and missed training until Thursday.
The Israeli midfielder will not travel but will be available for the midweek cup tie at QPR. He has been omitted from the squad since the home defeat by Arsenal in August.
Weaver was also tipped for a return soon, depending on the form of current no. 1, David Seaman, but his progress has been stalled once more.
"He finished a training session recently and then came in the next morning and had a pain in the knee," said City boss, Kevin Keegan. "It is a problem that we are going to have to live with as far as Nicky is concerned, but we will know more in a couple of weeks.
The problem is a reoccurence of the right-knee injury which kept him out for 16 months and required surgery from a specialist in the States.
"It is not great news and is a set back for him," Keegan added. "He has worked so hard to get back and at this stage we are still hopeful that in two or three weeks time he will improve and he can get back in training again."
"If it doesn't improve he will go back to America but that is long term and I don't want to alarm anyone at this stage."
Reserve keeper, Kevin Ellegaard, will take the step up after convincing his boss that he is ready to be called upon.
Keegan told the official City website: "I have every confidence in Kevin. Obviously he is a young lad so he lacks experience. You can say that about any young goalkeeper.
"But there are no problems from with his ability or with what we are seeing in training and what we have seen of him playing in the reserves."