Wenger's current deal is due to expire at the end of the season, and has refused to discuss his future in recent weeks.
But now having won his first piece of silverware in nine years, he has suggested that he is likely to sign a new contract, with Sky sources understanding an official announcement could come within days.
Asked if he was going to stay, Wenger told Sky Sports News: "Normally, yes of course."
And he admitted that the cup final win, a dramatic 3-2 success which came via an extra-time winner from Aaron Ramsey after they trailed 2-0, was a tough match.
"It was important, but it was very nervy and you could see that from the start but we recovered well," he said.
"Overall the spirit and togetherness of the team and quality saw us win the trophy which looked unlikely but that was down to Hull to who showed quality and that is credit to Steve Bruce who has done a great job with this team."
Wenger believes the triumph will be a huge boost to his squad and he believes it will be a 'turning point' for them.
"It was very important for this team and is a turning point to move on and it will take the pressure of their shoulders," he continued.
"We suffered in some games this season but this will give us something - belief and confidence, but overall we got 79 points, did well in the Champions League and we win the FA Cup and our team has shown remarkable consistency but we needed to win a trophy."