The Gunners haven’t qualified for the Champions League in the last two seasons under Arsene Wenger to culminate in the Frenchman’s exit.
Unai Emery replaced Wenger at the north London side and Arsenal have won their last four Premier League games to hoist themselves up the table.
The Gunners are in sixth spot in the Premier League table and a point adrift of Watford in fourth place.
Manchester United, meanwhile, were held to a 1-1 draw with promoted side Wolverhampton Wanderers in the Premier League at Old Trafford on Saturday.
The Red Devils have also lost Brighton and Hove Albion and Tottenham Hotspur to lose significant ground on Liverpool FC, Manchester City and Chelsea FC.
Former Liverpool FC defender Carragher believes Arsenal and Manchester United will ultimately miss out on a spot in the Champions League.
“On the evidence of what we’ve seen I’d say Arsenal or Manchester United [will miss out on the top four] from what we’ve seen in these opening six games,” Carragher told Sky Sports.
“The others look a little bit further up. Tottenham have had a little bit of a wobble, a good result for them on Saturday but it will be really tight and it always is with the top six.”
Arsenal lost to Manchester City and Chelsea FC in their opening two Premier League fixtures to make a disappointing start to the new campaign.
However, the Gunners were 2-0 winners against Everton on Sunday to extend their winning run to four games.
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