The north Londoners haven’t won the league since Arsene Wenger’s famous 'Invincibles' side in the 2003-04 campaign.
Manchester City and Liverpool dominated the title race this season, accruing 195 points between them. Pep Guardiola’s side ultimately claimed the trophy by just one point ahead of the Merseysiders on the final day of the season.
Chelsea were their nearest challengers a further 25 points adrift. Arsenal finished the campaign in fifth in Unai Emery’s first season in charge, a huge 28 points behind eventual champions City.
Despite that gap Van Persie believes his former club are capable of challenging for the Premier League next year.
"Of course they can, yes, because they're close already, they've been close for so many years," he told BBC Sport.
"It goes in periods. Now you have Manchester City, who are doing unbelievable. To win back-to-back titles is very special - that shows, really, that you're a proper team.
"But it goes in phases. Maybe [City manager] Pep Guardiola will leave next year and that does something to a club. Maybe Liverpool will take over and they will dominate for the next few years. But I hope and wish Arsenal all the best to win the title. I know how difficult it is."
The Gunners can still end the current campaign with a trophy when they take on Chelsea in the Europa League final in Baku next week.
Van Persie is again tipping his side to come out on top and claim their first European trophy since 1994.
"Personally, I think Arsenal have the edge," added the 35-year-old, who played for Arsenal for eight years between 2004 and 2012.
"They're a little bit better. But Chelsea have Eden Hazard and he is a very special player - at any time he can come up with something."