The Arsenal manager has come under increased pressure after a 3-1 home loss to Bayern Munich in the Champions League in midweek.
With the Gunners on the brink of Champions League elimination, a top-four Barclays Premier League finish is likely to offer their only hope of returning to the competition next season and they look set to go an eighth season without winning a trophy.
Arsenal have long been out of the title race and face an uphill battle to overhaul north London rivals Tottenham to get back into the top four.
The Gunners' healthy financial position means that it would not be a disaster in money terms for the club to miss out, but Wenger is refusing to consider that scenario.
"We want to be in the Champions League," the Frenchman said.
"Financially we can live with [not qualifying], but on the sporting front, no. And the sporting front is the priority."
FA Cup and Capital One Cup exits against lower-league opponents have also marred Arsenal's season, but Wenger, whose side hosts Aston Villa tomorrow, believes his players can regroup and have a strong end to the season.
Wenger is expecting a response from his team on Saturday, when full-back Bacary Sagna will miss out because of a knee problem, but defender Nacho Monreal is available again after being cup-tied in Europe.
"We know we are in a position where we cannot afford to drop points and we will go into the game with a huge desire to get out the disappointments that we recently had," he said.
"We want to come back in a strong position in the Premier League. I feel apart from Manchester United, it is very open for all the other positions.
"So it is down to us to continue to be consistent like we have been in the Premier League."
Despite Villa's troubles at the wrong end of the table, Wenger expects a tough test at the Emirates Stadium.
"We have to prepare ourselves for a battle," he said.
"They are fighting to avoid relegation and it is a typical Premier League game. There is no easy game for anybody."