The Lions have one more game on their 10-match tour, the series decider against the All Blacks at Eden Park on Saturday.
The 31-year-old New Zealand born centre complained of headaches before last week's match against the Wellington Hurricanes, dropped out of the squad to play the Super Rugby champions and had medical tests.
"While the findings of Jared's tests have been normal he still has a migraine and will not be fit to train until his symptoms have settled," Lions coach Warren Gatland said in a statement.
"The specialists are happy that the symptoms are not concussion related and he is expected to recover fully. We wish him a speedy recovery."
The Lions travel to Auckland later on Wednesday, having spent time relaxing in Queenstown ahead of the third test.
The visitors won the second game against New Zealand in Wellington 24-21 last Saturday, while the All Blacks won the first 30-15.
(Reporting by Greg Stutchbury; Editing by Ken Ferris)
