The 28-year-old was on the comeback trail from neck surgery and has not played for the All Blacks this year, but returned to training with them ahead of last week's Bledisloe Cup clash against Australia at Eden Park.
He was then released to play for Auckland in New Zealand's provincial competition for two matches before the World Cup squad is finalised on Aug. 30.
However, he was involved in a collision with loose forward Taleni Seu at training on Tuesday and knocked unconscious, the newspaper said.
"We were doing a defence drill and he had a collision with Taleni Seu," Auckland coach Paul Feeney said.
"They were going for the same ball and they both got concussed, it was just a freak accident five minutes from the end of the session."
The Auckland team were not immediately available for comment.
Faumuina underwent surgery towards the end of the Super Rugby season to relieve pressure on a pinched nerve in his neck that was causing him pain and diminishing his strength.
A strong scrummager, Faumuina is also well known for his ball handling and running in the open field, which would add to the attacking dimension of the All Blacks' forwards in England.
Faumuina, who has played 27 tests for the All Blacks since his debut in 2012, would miss the game against Taranaki on Sunday but was expected to be available for the match against Canterbury on Aug. 29, the newspaper added.
(Reporting by Greg Stutchbury; Editing by Peter Rutherford)
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