Oliver Croker was watching Bampton Football Club play Bradninch at the latter's Kensham Park ground on Saturday.
Club officials have said a hoop used as part of the fencing became lodged in his head, according to local paper the Exeter Express and Star.
A statement issued to local media by the club said: "As many will have heard, a freak accident took place at Kensham Park at the end of a football match between Bradninch and Bampton on Saturday 12 September.
"A hoop used as part of fencing became lodged in the head of a young lad from Bampton who had been watching the match.
"At this time, it is unclear how the accident occurred."
Another local paper, the Tiverton Gazette, said he was hurt when a post holding ropes that ran along the sidelines of the game struck him in the head after being uprooted.
The youngster was airlifted to the Royal Devon and Exeter Hospital with serious injuries after the accident.
He was then transferred to Bristol Children's Hospital but died overnight on Monday, police said.
A statement from the Devon and Cornwall force said: "A 12-year-old boy who sustained serious head injuries whilst spectating at a football match at Bradninch Football club on Saturday 12th September has died overnight.
"Our sympathy and thoughts are with the family."
The football club has released a statement saying: "Everyone at Bradninch Town FC would like to give our sincere condolences to the family of little Oliver who has tragically passed away.
"No one deserves to be taken so young and he will be dearly missed. Gone but never forgotten. Rest in peace."
A statement from his school, Tiverton High, said: "It is with great sadness that we hear of the death of Olly Croker in a freak accident at the weekend.
"In describing Olly, the first words that come to mind are that he was the life and soul of the party; a larger than life character, brimming with confidence, with an eagerness and enthusiasm for taking part in all aspects of school life.
"His youthful vibrancy and curiosity about life shaped his whole character.
"He was a very knowledgeable and sparky lad with his finger on the pulse of what was going on at school, in the local community and beyond.
"He was multi-talented and we had high hopes for his future. Olly was a keen sportsman and represented THS at rugby and even played up an age group.
"Our thoughts and wishes are with Olly's parents and brothers."