The former England striker believes the Dutch ace should beware the "grass is not always greener elsewhere" with the newly crowned Premier League champions ready to swoop
Woodcock called on the prolific Dutch striker to stay put after his future was left in doubt following a round of talks with the Arsenal hierachy over a new £130,000-per-week three-year deal.
Italian giants Juventus have ruled out a move for Van Persie, whose contract runs out at the end of next season, leaving the Premier League champions in pole position to sign the player this summer if he chooses to leave.
“I am sure there will be plenty of takers for him if he decides to leave Arsenal but one could do a lot worse I think by going somewhere else,” Woodcock told Goal.com.
“The problem is not so much him leaving English football it is if Manchester City come and knock on the door and then we are talking about loyalty.
“A lot of times people think that the grass is greener somewhere else and they go there and they find that does not happen so there are lot of things for Robin to take into consideration.
“And having said that, let’s be honest, Arsenal have been pretty loyal. Robin has one or two injury problems over the last few years and it would be interesting to see how many games he has played for Arsenal in the past and how many goals he has scored. You don’t have to plummet it all into one season. There is a bigger picture that you have to look at.”
The 28-year-old Dutchman’s 37 goals earned him the prestigious Professional Footballers’ Association and Football Writers’ Association player of the year awards but were not enough to end the Gunners’ trophy drought.
Despite that, Woodcock believes that Van Persie should stay at Arsenal because of the stability the club offers.
“Arsenal are one of the two most stable clubs in the Premier League along with Manchester United and you just have to look at how long their managers have been there to see that,” the former England international continued.
“You have to look at the other aspects. And I am sure he will be looking into that. It can’t be a case of just going for the money. I know some players will go if you offer them more because they just want to go for the money.
"But I would like to think that top players had an attitude that it is more about the game and where I am playing then just money and the contracts I am going to sign.
“I think Arsenal are in a reasonably good position and I think that Robin knows what Arsenal have got to offer,”
“If we go back through tradition right up to modern-day football they are a good stable club that have had some great players down the years.
"The only thing that has been lacking over the last few years is actually picking up some silverware but they are not too far off doing that.
“That is particularly so when Robin seems to have got over all his injury worries and has had a very good season so far and so I am hoping this is the starting point for a number of excellent seasons at Arsenal.”
Woodcock also insisted that silverware and not money will be the motivation for Van Persie as he plots his future.
He continued: “It is always important to keep any good player at your club and to be quite honest if you look at the bigger picture and the G14 group of the top European clubs those 14 clubs are going to be looking to win trophies and that is what every top footballer wants to be doing.
“He wants to be at a great club, he wants to have a very good contract and he wants to try to win some silverware down the line. So there are only a number of clubs that you can do to in order to improve. Are they on the list? Are they on the cards? That is one question. The second question is where do I want to progress myself.
“When you speak about players likes Robin van Persie and ones who are playing at the top of the Premier League it cannot be a money factor. If these guys have got their heads screwed on right then they will be financially secure so it can not be about earning a couple of million pounds more.
"It has to be about more other significant things. It has to be about the ethos of the club. It has to be about where do I want to live. It has to be about the history of the club. It has to be about the stability of the club.”