Regan, who was a regular in Brian Ashton's front row during the World Cup, has been granted a weekend off by Bristol, who play Leeds Carnegie in the Anglo-Welsh Cup on Sunday but have no hopes of qualifying for the last four.
But the 35-year-old has not been given permission to face the Springboks, with Bristol insisting he would be in breach of contract.
Premier Rugby, the body that runs the Premiership, can bar players from appearing in the game because it does not take place on a pre-agreed international weekend.
Bristol chief executive David White told BBC Sport: "Premier Rugby have made their decision and we are supporting that.
"We have not released him, he can't play and we have the jurisdiction to fine him.
"If he does play he'll be breaking his contract and will have to face the consequences."