The former West Ham United, Tottenham Hotspur and England midfielder, who joined Manchester United back in 2006, marked 11 seasons at Old Trafford and netted a long-range equaliser for a 2008 Red Devils side in a testimonial match on Sunday (4 June).
Speaking ahead of that entertaining clash at Old Trafford that saw Edwin van der Sar, Rio Ferdinand, Nemanja Vidic, Gary Neville, Paul Scholes and Ryan Giggs et al reunite to draw 2-2 with an All-Star XI featuring the likes of Jamie Carragher, John Terry, Clarence Seedorf and Michael Owen, Wenger congratulated Carrick on his achievements and expressed regret that he had never managed the frequently underrated 35-year-old.
"I give Michael my deepest congratulations for his testimonial," he told Arsenal.com. "I think he deserves it hugely.
"I would say personally I'm an admirer of his career, of his quality. One of my regrets is never having had him as a player because I think he was one of the greatest players in English football.
"I believe as well that he did not always get the reward and the tribute, the recognition he deserved, But I'm an admirer of Michael and I wish him all the best for the future."
A veteran of more than 450 appearances for United, Carrick's future had remained open to speculation with his previous one-year contract due to expire at the end of June. However, the club confirmed last weekend that he had agreed to prolong his stay by a further 12 months with Jose Mourinho glowing in his praise of a "fantastic human being and great role model".
Midfield stalwart Carrick has won a plethora of silverware since being brought to Manchester by Sir Alex Ferguson in an £18.6m ($24m) deal and inheriting the number 16 shirt vacated by Roy Keane, a haul that includes five Premier League titles, one FA Cup, three League Cups and now both major European club competitions. He featured 38 times - 30 as a starter - last term as Mourinho collected three trophies during his maiden campaign in charge.
While the ship has long since sailed with regards to any potential deal for Carrick, Wenger is expected to add to his Arsenal squad this summer after confirmation of that controversial two-year contract extension.
Pushed on his transfer plans during a recent interview with beIN SPORTS, the divisive Frenchman claimed that the Gunners, absent from Champions League competition next season for the first time in two decades, would make a maximum of two or three signings while warning that they cannot hope to compete with big-spending rivals that have "external resources that allow them to be basically unlimited".