According to a source at the club, Van Persie has put in the request after defender William Gallas was released upon the expiry of his contract.
Gallas wore the number 10 shirt for four seasons after moving from Chelsea to the Emirates Stadium in 2006.
The Dutch striker had previously been dissuaded from inheriting the number 10 for fear of causing undue comparisons with Gunners legend and fellow Holland international striker Dennis Bergkamp.
Bergkamp, who was inducted into English football's Hall of Fame in 2007, wore the number 10 for Arsenal over the course of a remarkable 11-year stay in north London.
It is thought Van Persie's number 11 shirt could be handed to new signing Marouane Chamakh.