The 31-year-old is reported to have accepted he may be demoted to a more peripheral role.
The report claims that Van Persie believes he has more to contribute to the side and he wants to keep working alongside Louis van Gaal.
Van Persie will enter the final year of his United contract next season but the club have a policy of offering one-year deals to players over the age of 30.
The deal would have worked out at roughly £220,000 per-week on a three-year term.
The Netherlands international was reported to have consulted Fenerbahçe winger Dirk Kuyt and Galatasaray playmaker Wesley Sneijder regarding a potential move.