The Gunners suffered back-to-back 3-0 defeats at the hands of Manchester City in less than a week – the first in the Carabao Cup final at Wembley on Sunday, 25 February, and the second during the league tie at the Emirates on Thursday, 1 March.
The north London club are sitting sixth in the table with 45 points after 28 games, trailing fourth-place Tottenham Hotspur by 10 points.
Things are looking quite grim at the moment, with Wenger's side knocked out of the FA Cup by Nottingham Forest in the third round of the competition earlier in January.
With 10 Premier League games remaining in the 2017/18 season, it is difficult for Arsenal to qualify for next season's Champions League by finishing in the top four. The other option which could help them seal a berth in Europe's elite club competition is quite difficult as well – winning the Europa League. However, even before the Gunners can entertain any thoughts of clinching the trophy, they will first need to overcome AC Milan in the Round of 16, which is by no means a simple task.
As for Wenger, the Frenchman signed a two-year deal last summer and insisted that he would honour his contract.
However, Carragher claims the only way Arsenal can end the negativity surrounding the club is by coming out and announcing that Wenger will step down after this campaign.
"Forget tonight, the only way you can turn this negative at Arsenal into a positive is if the board – together with the manager – comes out and announces that Wenger will move on in the summer," Carragher told Sky Sports.
"Everyone could get behind them. We'd then start saying, 'What a man – one of the great managers in the Premier League, one of the greatest managers in British football.'
"That's what should be being said but I think some Arsenal fans may even think they don't want to win the Europa League to get into the Champions League because then the manager may then stay and that's sad.
"I love Wenger, he's a legend, but I think the time has come now for the people above to say, 'Enough is enough – it's time for someone else to come in and take us in a different direction.'
"It's not that he's doing a bad job – he's won three FA Cups in the last four years and that's a good job – but someone needs to come and do a great job. People are questioning the manager and the players but they have to look at those in charge of the club."