The Netherlands international recently announced that he will not sign a new contract with the Premier League side, and has since been continually linked with a move away from the Emirates.
Clubs such as Manchester City, Manchester United and Juventus are all keen to lure Van Persie away from his current club, but if it was up to Koeman, the attacker would return to the club where he rose to prominence.
"Hey @persie_official, if Manchester United and Manchester City don't agree terms [with Arsenal], you can always return to Feyenoord. #winningtrophies," Koeman said on his Twitter account.
The 28-year-old joined the Feyenoord youth academy from Excelsior in 1998, and won the 2002 Uefa Cup with the Stadionclub before joining Arsenal in 2004.