"James continues to make excellent recovery from his head knock last week," said Lions doctor James Robson.
"He has done various grades of activity with no ill effect and he has done non-contact rugby activities with no ill effect. We have high hopes of him being passed tomorrow.
"He is subject to an independent neurological assessment subject to IRB regulations which takes the decision out of our hands."
Hook will visit the same neurosurgeon who ruled Ryan Jones was not fit enough to join the tour after suffering a blow to the head playing for Wales in North America.
If passed fit, Hook is likely to feature at Newlands on Tuesday and he could also come into the equation for the second Test.