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.
Turkey's NTV Spor reported Monday that Besiktas' basketball funds have been tied up by the match-fixing scandal that his ripped through the Turkish soccer world, forcing Besiktas to search for outside funding to help with the further pursuit of NBA players.
According to NTV Spor, Besiktas initially was prepared to offer Bryant a monthly salary of $500,000. But Bryant, according to the report, is seeking a monthly salary of $1 million to join Williams.
Sources told ESPN The Magazine's Chris Broussard earlier this month that Williams' overall deal with Besiktas is worth $5 million.
Sources say Williams would not be required to report to the Turkish club before the end of August or early September and that his deal with them would include an immediate out that allows him to return to the NBA as soon as the work stoppage ends.
Bryant, who has won five NBA titles with the Lakers and is a 13-time NBA All-Star, has been on a tour of China, including a Wednesday stop in Shanghai. He said he would consider playing overseas during the lockout and mentioned China and Turkey as possibilities.
Representatives of other NBA players also have contacted Besiktas, Ataman said, without identifying any of them.
The Turkish League season starts in October.
Besiktas last year signed Allen Iverson, the NBA's MVP in 2001, but his time in Turkey was cut short due to injury.
Another NBA player, 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 option for a second year.
Nets draft pick Bojan Bogdanovic also signed with Fenerbahce of the Turkish League in June.
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, which 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.
Information from ESPN.com's Marc Stein and The Associated Press was used in this report.