The Gunners have won their last four Premier League games to bounce back from a difficult start to the 2018-19 campaign.
Arsenal lost to defending Premier League champions Manchester City and bitter rivals Chelsea FC in their opening two fixtures.
The north London side have eased to victories over Newcastle United, Cardiff City, West Ham United and Everton to hoist themselves up the table.
Arsenal appointed Unai Emery as their new manager in the summer after Arsene Wenger left the north London side following 22 seasons.
The Spanish head coach signed five players in the summer transfer window to revamp the Gunners squad.
Former Liverpool FC captain Souness believes the Gunners will miss out on a Champions League spot in the third successive campaign.
Souness told Sky Sports: “Six games in, it’s very difficult but I’d go Liverpool or Man City, then Chelsea, Man Utd or Tottenham and Arsenal sixth. I don’t think Arsenal will get there [in the top four].”
The Gunners will welcome fourth-placed Watford in the Premier League at The Emirates next weekend.
Emery’s side are a point adrift of the Hornets in the race to secure a Champions League spot.
Arsenal haven’t won the Premier League title since 2003-04 when the Gunners finished the campaign unbeaten.
The north London side have won the FA Cup crown in three of the past four seasons.
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