The New Zealand striker recently penned a three-year contract with the Students following a stint with Bharat FC in India.
"We are not the only team affected by the Home Affairs rule but a host of other PSL teams are," Wits CEO Jose Ferreira tells snl24.com/kickoff.
"Chances of him [Bright] being available next week are very slim, if none."
Ferreira also confirmed the club has retained the services of veteran midfielder Dillon Shepherd for next season, while they have yet to make a decision on Lerato Chabangu.
Chabangu has been training under Gavin Hunt since his contract expired at Moroka Swallows at the end of June.
The Clever Boys will take on Matsatsantsa at Bidvest Stadium on Tuesday, with kick-off at 19h30.