Walcott chalked up more than 30 first-team appearances last season, and scored his first goal for the Gunners in the Carling Cup final.
With former captain Thierry Henry sold to Barcelona over the summer, Van Persie - back after five months out with a broken metatarsal in January - is likely to be seen as the main threat for the Gunners nest season.
However, van Persie firmly believes Walcott will be another to watch out for when the new campaign kicks off against Fulham this weekend.
"I have been practising with Theo since June 18 and I can assure you that he is up for it big time," van Persie told Arsenal.com.
"When we were doing tests over the summer it was very hard. We were running every day with the fitness coach and I could see it in his face he was really going for it."
Van Persie, who turned 24 earlier this week, feels Walcott's burning ambition can help him achieve great things with the Gunners.
"Theo is a young man - but he has big, big ambitions for himself to show his qualities," said the Dutchman.
"His ambition is one of the biggest things about him. He is just 18, but I see in him a big, big desire achieve his goals."