The Gunners have not won a competition since the 2005 FA Cup, and were held to a 2-2 draw by Swansea in their third round tie on Saturday.
They also face Bayern Munich in the last 16 of the Champions League and Wilshere believes his team-mates can beat their German opponents when the competition resumes in February.
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"We’re all hungry for it and maybe if we get a trophy this year it can lead to a few more.
"We've got a good crop of young players and experienced ones as well with [Mikel] Arteta and others, so if we get a trophy this year or in the next couple of years we can really build something, a strong team and be strong in the next few years.
"Obviously it is easy to sit here and say that when we get one trophy it will happen — but we need to get that first trophy.
"We are still in the FA Cup and the Champions League. I think we are in a good position, we just need to make sure we get the results now."
The Gunners sit 18 points behind league leaders Manchester United and face second-placed City on Sunday, with Wilshere lamenting how far off the pace Arsenal are in the Premier League.
He added: "I'm not sure if it's make or break but we know we need results. We're not in the position in the league that we want to be. We want to be higher.
"We've got a couple of games in hand and we have to make sure we make them count. There's no point having them and not getting results or making them count.
"We've got a tough few games and we know that in England this time of year is so important and we have to recover well and go again.
"We've got a big game on Sunday and we know City are a good team but we're at our home so we have to take the game to them, and then on Wednesday we have to do the same thing."