The fee was not disclosed but the BBC reported an initial fee of three million pounds would be paid, potentially rising to six million.
United's website said the 18-year-old had passed a medical and agreed a four-year contract starting on July 1.
"We have scouted Nick for months now. He is an exceptional talent and has been schooled well at Crewe. I'm delighted he has chosen to come to United," said manager Sir Alex Ferguson.
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