The fullback played his best football at Manchester United under Solskjaer and is currently a free agent.
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Williams was handed his senior debut by Solskjaer and also played for Norwich City and Ipswich Town, which cut him loose last season.
The defender was in May handed a 14-month suspended jail sentence for dangerous driving.
Williams, now 24, is eager to get his career back on track and The Sun says Solskjaer is willing to give the former England U21 fullback a fresh chance in Istanbul.