The Dutch international revealed his son has already shown some footballing promise and that scouts from the Arsenal youth set-up had invited him for training.
Despite this interest, Van Persie said he was not going to rush the five-year-old in to anything and was happy with him at his current club close to their home.
“I honestly have to say that he has got a lot of talent for football. I recognise a lot of stuff of myself in him,” he said.
“His coach is really great. Arsenal has invited him a couple of times to come and play there but he likes it at his club. He's sensitive for a good atmosphere. Just like me.”
The 28-year-old was speaking to reporters when he revealed his son also has a talent for golf – sometimes beating his father.
“He loves to play golf. You've got some great golf courses here. He's taking lessons and we regularly play a round,” he added.
“Sometimes he beats me! He's got a great gift for where the ball has to go. My shots sometimes end up in the bush; his always go straight.
“He says: 'Dad, I'm going to chip now, gimme my this or that club.' He can do that perfectly as well.”
The north London club may have to wait a while for Van Persie junior to make an impact, but they will be hoping his father is ready to make a decisive contribution this weekend.
Arsenal face second placed side Manchester United on Sunday at Emirates Stadium in what is a tough test following the disappointing loss to Swansea City last weekend.