Williams, a 24-year-old fullback, had a successful stint at Manchester United under Solskjaer's guidance.
However, he is currently a free agent after being released by his previous clubs, Norwich City and Ipswich Town.
Williams made his senior debut at Manchester United, which was a significant milestone in his career.
Despite facing challenges, including a 14-month suspended jail sentence for a traffic offense, Williams is eager to revive his football career.
Solskjaer is reportedly open to giving Williams a second chance at Besiktas, potentially paving the way for a fresh start in Istanbul.