The 20-year-old Amsterdam-born player received emergency treatment on the pitch during the match against Werder Bremen on Saturday.
He was airlifted to a hospital in Innsbruck, Austria, where he was put into an induced coma.
The club said a brain scan and other neurological tests had "not detected any anomalies" but that his brain function could not be accurately tested while he was being kept asleep.
On Thursday, the club tweeted to say it was "deeply saddened" to confirm Nouri "had been diagnosed with serious and permanent brain damage".
They added: "Our thoughts and prayers go out to him and his loved ones in this difficult time."
The club also released a statement, saying: "Nouri's brain functions have been further examined over the last few days at a hospital in Innsbruck.
"The diagnosis is that a large part of his brain is no longer functioning and that the chance of recovery of these crucial brain functions is nil.
"This is most likely due to insufficient oxygen supply to the brain after the midfielder became unwell during the Ajax - Werder Bremen match in Hippach.
"It is now expected Nouri will be taken to a hospital in Amsterdam in the near future for further care."
Earlier in the week the club said Nouri had experienced heart arrhythmia - an irregular heartbeat - during the match.
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