Once leaders, Arsenal have won just one of their past five Premier League matches to slip seven points behind table-topping Liverpool.
Wenger said the pressure to reach the Champions League had been great while the club were paying off the Emirates Stadium.
But the Frenchman, whose team meet fifth-placed Everton on Sunday, said while the financial pressure was no longer great, missing out would be a huge blow.
"Financially, the pressure is less. On a sporting side it is greater because we have been the leader of the league this season so not to make it ... the pressure is bigger," Wenger said.
"We want to be in there, but it's down to our results in our last six games to make sure that happens.
"Since we built the stadium, it has always been the obsession. I got that reproached enough.
"It was an obsession because of the financial compulsory situation. I’ve always been guided by - no matter what what happens - get in the Champions League.
"But for me it's not financial - it was before. What it is now is to play against the best – pride and prestige."
Wenger has described the trip to Goodison Park as a 'six-pointer, with Everton potentially moving within a point of Arsenal with a win and a game in hand.
The 64-year-old said his team's experience of finishing in the top four would be important late in the season.
"What plays a part is the memory that you have done it before, so we know we can do it," Wenger said.
"It is (a six-pointer). For us, it's an opportunity to go up forward and as well an opportunity to make a distance with Everton of course. It's both.
"As I told you after the last game against Man City - we look in front of us and behind us as well. That's what the game is about on Sunday.
"I think we have turned the corner and we have an opportunity to show that against Everton."
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