If successful, the proposed ban announced Monday would follow a similar measure enacted in Hawaii last year. Oregon and Washington are also considering similar legislation.
Shark fins are used to create a luxury Chinese soup that can sell for as much as $40 a bowl.
Supporters of the ban say shark finning is a cruel practice in which fishermen slice the shark's fin off while the animal is still alive and then throw the shark back in the sea to die.
Democratic Assemblyman Paul Fong, of Cupertino, introduced the legislation with fellow Democrat Jared Huffman, of San Rafael.