The Dutchman plundered 30 Barclays Premier League goals to help fire Arsenal to a third place finish, ending what had been a testing campaign on a high with Champions League football secure once again.
The 28-year-old still has 12 months left on his current deal, but so far talks over what is understood to be a lucrative extension have yet to be concluded.
Van Persie looks set to head out to the European Championships without making any firm decision, amid reports the likes of Manchester City and new Italian champions Juventus are ready to offer big money to lure him away from Emirates Stadium.
Former Arsenal defender Keown - who won three league championships during his nine years at Highbury - feels Van Persie should be allowed some space to consider what is likely to be the most important decision of his career.
"We need to give Robin time and not suffocate him," said Keown, who was speaking at the Carlsberg Pub Final, a national tournament to find the best Pub Cup team in England, which took place at Emirates Stadium.
"Robin understands and we all understand he is very much a special player for Arsenal. The adulation and respect which he has at Arsenal, Robin has worked a footballing lifetime to get that.
"If he leaves, he might find it hard to grow into another club with the respect he has here.
"However, he might feel he wants to go to a club where he thinks he can win things - that is probably the dilemma for him, rather than the financial one.
"It is swapping the love for him here if he goes to another club, he might find it a little bit of a difficult transition which he does not want to make."
Keown told Press Association Sport: "It is his heart against his head.
"It would be quite refreshing if a player decides: 'This is my club, I am not going to play anywhere else and am going to show that faith and trust which was shown in me. I am going to believe in project Arsenal have'.
"We will have to see what answer he comes up with, but of course Arsenal are desperate for him to stay and have lost too many players in the past.
"If Van Persie were to go as well, it would take them a would take them a number of years for them to recover from that."
Arsenal overcame an appalling start which saw them lose four from the opening seven league games to overhaul rivals Tottenham into third place.
Keown feels should Arsene Wenger's men be able to recapture the form of the latter part of the campaign, then 2012/2013 could yet turn out to be a vintage campaign.
He said: "It ended up as a dream finish for Arsenal, because at the start of the season, it was a bit of a nightmare in terms of results.
"Arsene Wenger will now look to improve things for next season, when it will just be about consistency."
Keown was impressed by the efforts in the Carlsberg Pub Cup final, which was won by south London side International FC, representing the Britannia pub.
They will now head out to Euro 2012 where the side will be managed by former England international Gareth Southgate to take on a team of European all-stars, including former Liverpool midfielder Patrik Berger and Chelsea winger Bolo Zenden, ahead of attending the England versus Sweden game in Kiev.
Keown said: "The Carlsberg Pub Cup finals have been played at a very good level - and there are one or two you wonder why they are not playing at clubs."