The teenager was taken to hospital and the club confirmed on Sunday that he had later died.
As the national men’s team posed for a photo ahead of their World Cup qualifier against Poland in Warsaw, Kane held up a shirt with print reading “For Dylan”.
A post on the England team’s Twitter account featured the photo and read: “Rest in peace, Dylan Rich. Forever in our thoughts.”
West Bridgford Colts said they were “devastated and heartbroken” by Dylan’s death.
A statement from the club read: “It is with great sadness that, following the events of Thursday evening, we have to announce the tragic passing of Dylan Rich.
“This news has left everyone involved with our club devastated and heartbroken.
“Our thoughts and prayers go out to his parents Mike & Anna his sister Lucy & Dylan’s family, friends and teammates.”
The match was abandoned as a result.
The FA said it was “deeply saddened” by the “tragic news”.
A statement read: “We send our deepest condolences to his family, friends and everybody at the club at this very difficult time.
“We would like to recognise the courageous response of all those who reacted quickly to the situation at the FA Youth Club fixture on Thursday evening and we thank the emergency services for their efforts.
“We ask that the privacy of both his family and the club is respected at this time.”
Football clubs such as Nottingham Forest, Notts County and Boston United also shared their condolences on social media.
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