According to the Daily Star's David Woods, Wenger has been impressed by the Dutchman and will have a say in who succeeds him at the Emirates Stadium, albeit he will not be solely responsible for the decision.
The 42-year-old earned over 100 caps for the Netherlands and played under Wenger at Arsenal in a career that also saw him enjoy successful stints at Rangers, Barcelona and Feyenoord.
Van Bronckhorst took over the Dutch club in 2015 and led them to the KNVB Cup in his first campaign, before guiding them to their first Eredivisie title since 1999 last season, in which his side finished the year with a positive goal difference of 61.
Wenger's future was the subject of much speculation last season as the Frenchman was in the final year of his deal.
However, as Bleacher Report's James McNicholas noted, the club had done little to prepare for his departure:
Back in 2014, many assumed AW was signing his last contract. Yet it's astonishing how little succession planning has happened since then.2017-3-5 11:20:52
Despite that, a vocal section of the Emirates fans were eager to see the manager call time on his career at Arsenal, per The Sun's Neil Ashton:
neil ashton @neilashton_
"Arsene Wenger, you're killing our club". Ouch. https://t.co/vfWuLF5mvY2017-3-7 18:56:34
Wenger even said he would take fan feeling into consideration in regards to his future, according to ESPN FC's Mattias Karen:
Mattias Karén @MattiasKaren
Asked if fan sentiment will affect his decision about his future, Wenger clearly states: "Yes."2017-3-9 09:12:46
However, he signed a two-year extension on his deal, and while there is a lot of football left to play this season, the club are again well off the pace in the Premier League as they lie in sixth, 12 points behind leaders Manchester City.
Having a succession plan in place for his eventual departure is critical, though, as it will be no easy task to replace a man who has served as manager for 21 years so far.
Van Bronckhorst is a promising young boss with experience of Arsenal, and the title win he oversaw last season will have enhanced his prospects of one day getting a chance at an even bigger club.
Gunners supporters may prefer a more experienced coach with a track record of success in a stronger league or Europe, though, and appointing someone of that calibre would perhaps be less risky.
Either way, it is a decision Arsenal must get right.
Manchester United are still recovering from the decision to appoint David Moyes and then Louis van Gaal after Sir Alex Ferguson retired, and they were in a stronger position when he left—having just won the title—than Arsenal are now, so the impact of a similar mistake could be even more devastating.