The Gunners head to Swansea on Sunday for their FA Cup third-round tie looking to keep alive hopes of delivering a first piece of silverware to the Emirates Stadium.
While Wenger knows the importance of continued Champions League qualification, the French coach insists he has not lost the taste for an open-top bus parade.
"Of course it bothers me because I am not here to watch other people win trophies," said Wenger, whose side last tasted glory when they lifted the FA Cup in 2005.
"It is important and I will give my best to win on Sunday. I will just remind you that I have won the FA Cup four times.
"Of course that [reminders of a trophy drought] is not pleasant, but the only way to do it is to win."
Wenger, however, accepts any talk of Wembley would be premature, even if his side progress from what is a tough encounter in south Wales.
"It is the first round (we play). You cannot speak about winning a trophy, so for us it is important to show continuity and consistency, and the FA Cup is part of that," said Wenger.
"To get people to believe more in the team, it is important that we win on Sunday.
"It is an important moment of the season for us, for the team to strengthen the confidence of the team.
"We feel we are on a good run, but how far can we go? To win in the FA Cup will strengthen [our belief]."
Arsenal have produced some form in the Premier League over the past few weeks, their 1-1 draw at Southampton halting a run of four straight wins.
Swansea, though, were the last team to record a victory at Ashburton Grove, when they ran out 2-0 winners on December 1 following a late brace by Spanish striker Michu.
Wenger challenged his side to set the record straight when they travel to Liberty Stadium.
"They had a very good game here. We have to give them a different response on Sunday," Wenger said.
The Gunners are expected to make some moves to bolster the squad this month, with a £10million bid for Atletico Madrid striker Adrian Lopez said to be on the table, while fringe players like Johan Djourou and Sebastien Squillaci will be off-loaded. Morocco forward Marouane Chamakh has already joined West Ham on loan until the end of the season.
However, Wenger wants concentration to remain on the side producing the required performance against Swansea rather than thinking about how to boost any perceived deficiencies.
"The most draining is that you get the best out of your team and worry about the next game. In our job it is important that you focus on what is really important, the players who can help the club to win the games - those who are somewhere on the transfer market will not help you to win the games," Wenger told a press conference.
"For us it is important to focus on our next game and our next game can only be won by the players who are here."