Ataman said Bryant was waiting for an offer from Besiktas. The club, however, said it would need a sponsor to be able to pay for Bryant's contract.
"Our board will evaluate that," Ataman said.
Bryant has won five NBA titles with the Lakers and is a 13-time NBA All Star.
Representatives of some other NBA players have also contacted Besiktas, Ataman said, without naming any of them.
The Turkish League season starts in October.
Williams averaged 20.1 points and 10.3 assists for the Nets and the Utah Jazz last season. Besiktas signed Allen Iverson, the NBA's MVP in 2001, last year but his time in Turkey was cut short due to injury.
Another NBA player, New Jersey Nets guard Sasha Vujacic, signed last week with Turkish club Anadolu Efes. The 27-year old Slovenian guard agreed to a one-year deal with an optional second year.
Talks between the NBA and the union broke down June 30, and the owners decided to impose the league's first lockout since the 1998-99 season was shortened to 50 games. Both sides say there are significant differences between their proposals, raising the possibility that all or part of next season could be canceled.