Van Persie has been linked with a move away from the Emirates Stadium as he has just 12 months left to run on his current contract.
Arsene Wenger does not want to see his skipper and star striker depart, but the club risk Van Persie leaving for nothing when his deal expires next summer if he does not agree a new one in the meantime.
Manchester City have been strongly linked with a move for the Holland international, while Real Madrid and Barcelona have been mooted as other possible destinations.
Arsenal chief executive Ivan Gazidis confirmed discussions had taken place at the end of the season and insisted the north London club would not be rushed into making any announcement on Van Persie's future.
Now the striker's father, Bob, has ruled out the possibility of a switch to the Etihad and also moved to quell suggestions his son could move to Spain.
"He'd never go to Manchester City as it's impossible he would play for another Premier League side that wasn't Arsenal," Bob van Persie is quoted as saying by El Mundo Deportivo.
"It would make no sense to go to Barcelona when they've already got Leo Messi.
"Real Madrid would be a good option, but they are not so much a team as a collection of star names, so he wouldn't really fit in there either."
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The 2012/13 Arsenal away replica soccer jersey. Youth Available
The 2012/13 Arsenal replica soccer jersey. Youth Available