The 31-year-old Netherlands international had the first part of his medical in England on Thursday and is set to a sign a four-year contract with Fenerbahce, the clubs having reportedly agreed a fee.
The Turkish side announced on their Twitter page: ???Fenerbahce has begun transfer talks with Robin van Persie and his club Manchester United.???
Van Persie has a year left on his United contract but now looks set to link up once again with his former Old Trafford team-mate Nani.
Despite having struggled with injury at times, Van Persie has been one of the most prolific goalscorers in the Premier League since joining Arsenal from Feyenoord in 2004.
He moved on to Manchester United for ??24m in the summer of 2012 and played a key part as they won the Premier League title in his first season at Old Trafford, winning the Golden Boot which is awarded to the top-flight???s leading scorer.
However, his subsequent two seasons with United have been far less successful and he lost his first-team place at times last season, despite the arrival of his former national team boss Louis van Gaal.
The Fenerbahce fans greeted Nani ecstatically when he arrived in Turkey and Van Persie is likely to receive a similar welcome this weekend.