The 67-year-old also became the FA Cup's most successful manager as he claimed a seventh winners' medal.
After finishing fifth, Wenger and Arsenal missed out on Champions League football for the first time in 19 years.
That came on the back of a 10-2 aggregate defeat to Bayern Munich in this season's competition, with some sections of supporters calling on Wenger to leave this summer, rather than sign fresh terms when his current contract expires next month.
Arsenal manager Arsene Wenger saw his team miss out on a Champions League place
Image credit: PA Sport
The Frenchman has yet to announce publicly whether he will be manager at the Emirates Stadium next season but did admit his future would become clear following a pre-arranged meeting.
"We have a board meeting on Tuesday, and on Wednesday or Thursday it will all be very clear," he said.
Asked if delivering the FA Cup would be a perfect way to say goodbye having been at the club nearly 21 years, Wenger replied: "There is no perfect way.
"I just want to do well for this club and apart from that after that it is down to the board members am I the right man to take this club further and for me to decide if I'm the right man, that is it.
Arsenal's French manager Arsene Wenger lifts the FA Cup trophy after their win over Chelsea in the English FA Cup final football match between Arsenal and Chelsea at Wembley stadium in London on May 27, 2017. Aaron Ramsey scored a 79th-minute header to ea
Image credit: Getty Images
"It is not about popularity, it is about confidence. It would be a bit ridiculous that 20 years depends on one games and the future of the club depends on one games, we will see more next week."
Ahead of the final, Wenger revealed he has never kept a winners' medal in his managerial career.
But will be holding on to this one having moved clear as the most successful boss in the storied history of the FA Cup, eclipsing George Ramsay's six wins with Aston Villa between 1887 and 1920.
"For once I have kept my medal so that means it is a special night for me," he said.
"I'm very proud, if nobody has done it since 18-god knows-when it means it is not easy.
Arsenal's players celebrate after their win over Chelsea on the pitch after the English FA Cup final football match between Arsenal and Chelsea at Wembley stadium in London on May 27, 2017. Aaron Ramsey scored a 79th-minute header to earn Arsenal a stunni
Image credit: Getty Images
"I'm especially proud of the way we won this season, in the semi-final we had to play against Man City here and in the final Chelsea here, you cannot get tougher and we had two outstanding performances, this cup was won with special spirit and special attitude."
Wenger was once again critical of those Arsenal fans who have demonstrating against him during matches this season.
Planes carrying anti-Wenger slogans, boycotts of home games and signs and banners being displayed inside stadiums came during a blip for Arsenal which Wenger believes cost them dear.
"I'm in a public job, I accept to be criticised and people don't agree with me," he said.
"But once the game starts and you are a fan, you stand behind the team, that is what I have not accepted this season and I will never accept it. Outside of the game I never had any word with any journalist in 20 years when they criticised my decisions and it is the same with the fans. But when a game starts you support your team. We have played some games in a hostile environment and I never accept that."
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