The Associated Press, citing a person familiar with the injury, reported that Bush has a broken right fibula.
Bush was expected to be examined further Tuesday, according to the person who spoke to The AP on condition of anonymity because the team hasn't officially updated Bush's condition.
Betts, who rushed for a career-best 1,154 yards in 2006, has worked out for the Saints twice before and briefly was with the team during training camp.
The bone in Bush's lower right leg was broken as he recovered his muffed punt with 6:58 remaining in the game. He had trouble catching the ball at windy Candlestick Park and was injured during the scrum after he dove for the ball and players landed on him.
"Last second, it kind of shifted a little bit, just enough for me to drop it," Bush said. "I tried to recover it, and somebody took my leg out. I don't know who it was, what happened or how it happened. I just know my leg got taken out. That's it.
"It hurt like hell," Bush added. "When I got back up and tried to walk on it, it felt terrible."
The injury capped a difficult past few days for Bush, who gave back his Heisman Trophy last Tuesday. The NCAA punished USC after concluding that Bush and his family took money and gifts from sports agents while the running back played for the Trojans. Bush said he returned the award to end the controversy.
"It's been a pretty tough week," Bush said. "It's just the way life goes.
"I'm just trying not to think the worst and just try to stay as positive as I can in this situation. That's all you can do, hope for the best."
Bush had played well before the injury. He caught a 6-yard touchdown pass on New Orleans' opening drive and set up a field goal with a 43-yard punt return on the final play of the third quarter.
Bush finished with 30 yards on four receptions and 4 yards on five carries.
The Associated Press contributed to this report.