Dutch striker says he's impressed by the 'fantastic' 18-year-old and he could 'definitely' make the grade at international level, but fans must be patient with the winger
The youngster shone in the team's 7-1 win over Blackburn, scoring his first and second Premier League goals in the process.
"He is a fantastic player already; he's only 18 and is so young," Van Persie told the club's official website.
"He's good on the ball - he has everything. He's quick, he has movement, technique, can score a goal or give an assist. He has a great future."
But the Dutchman hopes that people are not too quick to apply the pressure to the player, saying that he will have his fair share of ups and downs.
"But don't put too much pressure on him. You always have periods when you are young when you have really high highs and you go up and down until you balance it a bit better.
"He will have ups and he will have periods when he's not playing so well. So my advice is to be aware of it and be patient."
There had been suggestion in recent weeks that Oxlade-Chamberlain could be in line for a call-up to the senior England team ahead of Euro 2012 after some impressive performances in the Under-21s, and Van Persie agrees that there is a big future ahead for the winger.
"Why not? In the future, definitely. He's a big prospect. Everyone is enjoying watching him. I do, you do and he brings something extra to the game and people just love it.
"It reminds me of my own period when I was breaking through and was 18 or 19. I believe when you have all these qualities it gives you confidence and you can enjoy yourself.
"He's not someone who thinks about negatives, just positives and you can enjoy the game like that.