The midfielder has been a free agent since leaving Liverpool at the end of last season.
According to The Times journalist Paul Joyce, Oxlade-Chamberlain has agreed terms with Besiktas.
He flew into Istanbul Airport on Tuesday ahead of talks with the club over a move.
The report adds that the 29-year-old will have a medical in the coming days.
Oxlade-Chamberlain has a worrying injury history across his time at Arsenal and Liverpool.
He missed 11 months after rupturing his anterior cruciate ligament in his first season at Anfield.
Knee, ankle and hamstring problems continued to disrupt his time at Liverpool, including last season.
The former England international missed the first three months of the season with a hamstring injury and was limited to just 13 appearances across all competitions.
Besiktas have held interest in Oxlade-Chamberlain for a number of weeks and first lodged an approach earlier this month.
Several Premier League clubs had also been linked with a move earlier in the window.
Oxlade-Chamberlain will have the chance to play in Europe if he completes his move.
Following a third place finish last season, Besiktas are competing in the Europa Conference League this term.
