The 33-year-old, England's leading test wicket taker, missed the fourth match at Trent Bridge which the hosts won by an innings and 78 runs to take an unassailable 3-1 lead in the series.
"Unfortunately, he's not going to make it," Cook told reporters. "He's gutted about it. He's made really good progress from where he was at Edgbaston but this is a week or so too soon.
"He bowled in the nets yesterday but at about 60-70 percent and that's not how you go into a game. The medical team have been brilliant with him but he's still about a week away."
Mark Wood is likely to keep his place in the team in a four-man pace attack with Stuart Broad, Steven Finn and Ben Stokes, although England could be tempted to play leg-spinner Adil Rashid on Thursday.
He could be a vital weapon in England's next series against Pakistan in the United Arab Emirates but Cook said the priority was to win the Ashes 4-1.
"We have come into this game trying to win the series 4-1," he said. "That is what we discussed.
"Clearly, Adil has been in every single squad. The pitches have not quite suited two spinners so that is the reason he has not played.
"Adil will have to wait his chance but I think he is really improving as a cricketer and I think he feels a lot more comfortable in the England environment. He will probably get his chance in Abu Dhabi."
The fifth test starts on Thursday.
(Reporting by Ed Osmond; editing by Amlan Chakraborty and Pritha Sarkar)
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