The Frenchman became the first manager to win the FA Cup seven times as his side beat league champion Chelsea 2-1, BBC reported.
The 67-year-old's contract expires this summer, and there has been much speculation about whether he will stay.
"I have kept my medal, so that means it is a special night," he said. "For the rest, we will see what happens."
Wenger said he knew what he wanted to do, but added, "Let us enjoy this win and not worry about the future."
He told the BBC: "We have a board meeting on Tuesday, and on Wednesday or Thursday it will be very clear."
It is understood Tuesday's gathering is not a formal board meeting, but Wenger's future will be discussed by the club's directors.
Alexis Sanchez gave Arsenal the lead against Chelsea, and Aaron Ramsey clinched the victory after Diego Costa's equalizer.
Arsenal won the FA Cup for a record 13th time as Wenger, who joined the club in 1996, became the most successful manager in the competition's history.
But Arsenal finished fifth in the Premier League, meaning it failed to qualify for the Champions League for the first time in 20 years.
And the Gunners' season has been beset by protests from supporters, demanding the Frenchman ends his long association with the club.
In an interview with Football Focus before the final, Wenger described the criticism he has had this season as "a disgrace" he will "never forget".
"I try to serve this club with always the same commitment and quality," said Wenger after the final.
"We had some difficult moments but I spent every minute of my season on full commitment and we never give up and that is why today I am more proud than usual because the team and myself refused to give up and we finished the job in a positive way.
"Nobody gave us a chance and we responded with attitude and class and we have seen that today."
With Laurent Koscielny suspended, and Gabriel and Shkodran Mustafi injured, Per Mertesacker made his first start of the season at Wembley.
A World Cup winner with 104 caps, the 32-year-old has had an injury-plagued season, playing just 37 minutes of first team football before Saturday.
But the German defender kept Chelsea striker Costa quiet for much of the game.
Wenger said, "Per Mertesacker, for me, is a perfect example for any young professional football player.
"What he did was the consequence of an unbelievable attitude every day even when he was not selected.
"When he was not selected he worked even harder and that is why I will pay a special tribute to him."
Asked about his manager's achievements, Mertesacker said: "Not bad, is it?
"That is everything I have to say because there has been a lot said about him and I think the team said everything they needed to."
Midfielder Ramsey had also scored the winning goal in the 2014 victory over Hull City.
Asked about Wenger, the Welshman said: "Of course, I want him to stay. He deserves this."
Victory brought striker Danny Welbeck his first piece of silverware since joining from Manchester United in 2014.
The England international said, "The manager is his own man and he makes his own decision and the board will make the right decision."
Former Arsenal striker Ian Wright, who scored 185 goals for the north London club and won the FA Cup in 1993 and 1998, told BBC TV: "You cannot read him. You cannot read what Arsene Wenger is going to do.
"What he has done is put a lot of faith in his team. In certain games, they have shown what they are capable of, and today was one of those days."