A stress fracture in the seamer's right shoulder hampered the 34-year-old's season in the summer and also forced him out of the current series in Bangladesh.
Anderson recently claimed that he was hopeful of returning to action in India, but he is unlikely to be part of the squad for the opening clash in Rajkot on November 9.
"I don't think he will be there for the start," Cook told reporters. "He might come out to do some training but he will not be ready to play in the first Test. They are making some decisions over the next seven days.
"I spoke to him last night. He has been doing a lot of training and is in good shape physically. The decision is going to be made over the next week on when he goes to India."
When asked if he expects Anderson to return at some point during the five-match series, Cook said: "Yes, I think so, but you need to speak to the medical team.
"Jimmy has not bowled yet and that has been the problem in the past. The test for his body is when he starts bowling but he is making some strides."
Meanwhile, Cook is on the verge of breaking Alec Stewart's record of 133 Test caps for England.