The Calif. state Assembly has already approved AB 376.
But before the full Senate can consider the bill, it must be passed out of the Senate Natural Resources Committee.
Jessica Lass with the Natural Resources Defense Council said getting the needed votes may come down to one or two votes.
"It looks like we just need one more vote particularly from Senator Padilla in Los Angeles and we're really kind of pushing him to make the right decision on this," said Lass.
Lass said the legislation would also ban the distribution of shark fins, which are used for soup.
"Millions, up to 73 million sharks are killed every year just for their fins," said Lass. "And there's a pretty lucrative trade within the U.S."
Lass said Washington state and Hawaii have recently passed similar legislation.
Dozens of actors and other celebrities have been active in their support of the California legislation.