England international Wilshere, who spent the previous campaign on loan at Bournemouth, has been plagued by injuries over the past few seasons and there were doubts over his future during the summer transfer window.
West Ham United were said to be showing an interest in the midfielder, who has been at the Gunners since 2001, while there were also reports claiming a switch to Italy was on the cards.
The 25-year-old opted to stay at the Emirates and he produced an impressive performance in the 4-2 Europa League victory against Bate at the Borisov-Arena on Thursday after returning to full fitness.
Wilshere, whose current deal expires at the end of the season, says he is happy to be back challenging for a first-team spot and he would like to extend his contract.
He told the London Evening Standard: "You are always playing for your future but I am happy to be back, to feel part of the squad. I am fit and healthy and getting back to my best and when the time comes to sit down and talk or not we will see.
"Do I see myself staying? Of course I do. I love this club. They have been good to me over the years, I have a great relationship with the boss.
"He [Arsene Wenger] has played me since I was 17. He has put his trust in me since then. We have a great understanding and of course I want to stay."