The Gunners handed their under-fire Frenchman a new three-year deal in the summer despite missing out on a top-four place.
Arsenal were 2-1 winners against defending Premier League champions Chelsea in the FA Cup final at Wembley.
The north London side have won the competition in three of the past four seasons under Wenger.
Arsenal haven’t won the Premier League title since 2003-04 when the Gunners finished the season unbeaten.
Wenger has won the top-flight crown four times but has never lifted the Champions League after losing to Barcelona in the final in 2006.
Atletico Madrid boss Diego Simeone has been talked up as a potential replacement for Wenger in the Emirates Stadium hot seat.
Bournemouth manager Eddie Howe is another head coach who has been touted as a possible successor for Wenger.
Former Arsenal midfielder Parlour warned the Gunners board to forget about unearthing another ‘Wenger’.
Parlour told Newstalk: “Arsenal have to start planning for the future. They won’t get another Wenger staying at a club for 20 years. They’ll stay three or four seasons at a maximum.”
Arsenal lost 3-1 to Premier League leaders Manchester City in their last game before the international break but returned to winning ways on Saturday with a 2-0 win against bitter rivals Tottenham Hotspur.
The Gunners haven’t won the top-flight crown since 2004 but finished in second place in 2016.
Arsenal spent a club-record £52m fee on France international Alexandre Lacazette in the summer transfer window.
The Gunners signed Bosnian defender Sead Kolasinac in a free transfer from Bundesliga outfit Schalke 04.
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