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 Barclays 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."
"People try always to create pressure around every game, but that is part of our job. We want to just win our next game," he said.
"We are on a good run and I hope that Swansea will face a different Arsenal this time."
He said: "Swansea have done excellent work, with good technical players and which suit the way they want to play.
"All of their buys have been intelligent buys."
Champions City come to Emirates Stadium in a week's time, which is followed up by a trip to Stamford Bridge and both games are set to fully test the character of Wenger's side.
"The best way to prepare for our games against Manchester City and Chelsea is to come out on top against Swansea, which will be a difficult tie," Wenger told Arsenal Player.
"The FA Cup is always important, but we are on a good run overall and we want to keep that going."
Wenger added: "It (the Premier League) is tight, every game is massive which is why consistency is so important.
"All the teams are difficult to beat and if you are not completely at your best, it does not work."
Wenger, meanwhile, hinted he could yet resurrect a loan move for former captain Thierry Henry during the Major League Soccer closed-season - although any imminent short-term deal with New York Red Bulls appears unlikely.
"At the moment it is definitely (off), but if we have big injuries and Thierry wants to do it... (however) we have not gone as far (as that) because we look more for permanent people," Wenger said.
"At the moment we have (Olivier) Giroud available, (Theo) Walcott available, we are not too short in this department, so we will see."