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The pacer had already been ruled out of the ongoing Bangladesh tour due to a stress fracture in his left shoulder.
He sustained the injury during play against Sri Lanka this summer, and missed the first Test against Pakistan in July.
"He won't be ready for the first Test match. He might be ready to do some training, but he won't be ready for the first Test. A decision will be made on when he comes out over the next week or so," Cook was quoted as saying by ESPNCricinfo here on Wednesday.
"I spoke to him last night and he is training well and in good shape physically."
"But he hasn't yet bowled and that has been the problem in the past. However physically well he's been in the rest of his body, it's when he starts bowling (that the problem occurs)," he added.
Another fast bowler Mark Wood is also likely to miss the India series with an ankle injury.