As a free agent, Walker-Peters is currently without a club after his contract with Southampton expired.
The 28-year-old defender has been linked to several teams, including Leeds United.
However, according to sources, Walker-Peters is in Istanbul for a medical examination and to discuss contract terms with Besiktas.
Walker-Peters had chosen not to sign a new contract with Southampton after the team's relegation from the top division last season.
He has been playing for Southampton since 2020.