However, the 24-year-old Ajax captain looks set to receive offers from many of Europe's elite in the next transfer window, with many scouts having looked at the Belgium international in the game at Old Trafford.
And it's the Gunners who should make the best effort to sign Vertonghen.
Wenger looks set to spend big this summer, with money available and fans, players and journalists alike clamouring for the club to splash the cash they clearly have and purchase high-profile stars to boost their ability on the pitch, and their brand equity off it.
Website Transfermarkt, used frequently in football to see a player's true transfer market value, values the Belgian at £10.5 million, and realistically it won't take much more than that to sign him.
It's a fee that would certainly keep Ajax happy, and is one which would provide great value for Arsenal without taking out a huge chunk of their transfer kitty.
Do Arsenal Need Jan Vertonghen?
Rumour has it the Gunners have a £57 million transfer budget for the summer, so a swoop for Vertonghen should at least leave them with enough to sign an experienced, top-quality striker to help or replace Robin Van Persie, as well as perhaps a high-class playmaker.
First and foremost, Vertonghen should be snapped up by Arsenal though, before they look at strengthening in other areas.
The tall, imposing defender possesses the qualities currently lacking in the heart of Arsenal's defence.
For a start, his aerial ability is unrivalled in comparison the current Gunners' backline.
Vertonghen isn't the sort of static jumper who seeks to clear the ball and be careful not to give away a foul.
He's the kind of aggressive defender who'll dynamically launch himself into an aerial challenge to get maximum power on his header, regardless of the consequences.
It's the sort of domineering aerial play Thomas Vermaelen seems to be losing in his game, while Per Mertesacker and Laurent Koscielny never had it in the first place.
In other European leagues, his aerial style would have the same effect as the styles the current Arsenal defenders have.
But in the Premier League, his dynamic style gives him the aerial intimidation factor to out-muscle a lot of players—and it was clear in both legs against Manchester United in the Europa League.
Not only does Vertonghen have the strength to succeed as a centre-back in England's top-flight, he also has that intimidating style about him.
It'd make him more of a success in England than anywhere else as less fouls are given as a direct result of player tussles.
Vertonghen's no-nonsense leadership is also an undoubted quality which is sure to have interested many managers.
As is evident on the pitch, the Belgian genuinely hates making mistakes, and despises seeing his teammates commit errors.
Unlike most players in football, including every single member of the Arsenal team, when a teammate makes a mistake, Vertonghen doesn't just gesticulate his disdain, he has the initiative to tell that colleague directly.
And not just tell them, but really shout it at them, irrespective of how it may look to the opposition or outsiders.
Few players in football are that confident to do such a thing, mainly because they don't want to cause a rift with a teammate, or are scared their manager will punish them for trying to do their job for them.
Vertonghen on the other hand doesn't seem to care, and the way he shows his disdain for errors can often lift his teammates, showing that his team are a side who shouldn't make mistakes, and telling the opposition that they won't be so lucky next time.
When Arsenal concede any sort of goal, that sort of leadership in the defence is very rarely present.
Most often the players look at each other, bemoan their luck and just try to forget about it—there's no leader punishing a player and firing him up, as well as his teammates, not to concede again.
It's those two qualities, his leadership and his intimidating aerial style, which really set Jan Vertonghen apart from most other centre-backs.
On a technical level he's actually on a par with the Gunners' backline, but has the strength and "human aspects" so to speak that very few world-class defenders in football have.
He possesses the two main qualities that the whole Arsenal team lacks, and would actually come to North London as something the entire side, barring Bacary Sagna, is not—an established winner.
No Arsenal player has ever won a major trophy before, besides Sagna who won the Coupe de France with Auxerre in 2005.
Vertonghen, as then vice-captain, won the Eredivise title last season, and the season before lifted the KNVB Cup (Holland's equivalent to the FA Cup).
So out of any team in Europe, the club who would get the maximum benefit from signing the domineering centre-back is Arsenal.
An experienced professional with over 200 top-flight games under his belt, and a player still with potential and an excellent re-sale value at 24 years old, Jan Vertonghen represents the perfect signing for the Gunners.
An Arsenal team with the Belgian would strive harder to commit less defensive mistakes, would have a greater winning mentality and would also have a meaner defence which would be much harder to out-muscle.
And that, more than anything else, is the first thing any team needs to be a successful side in football.