The Gunners crashed out in the semi-final's with a drab 0-0 draw to Villarreal, losing 2-1 on aggregate, as their old manager Unai Emery came back to haunt them.
There have been calls for Arteta to go with the club destined to be without European football for the first time in 25 years.
One unhappy Gooner even told The Sports Bar the Spaniard should be replaced by Arsene Wenger until the end of the season.
If he does go, the rumour mill will go into overdrive over who the club should choose to replace him.
Massimiliano Allegri, the former Juventus boss, is the current frontrunner as the next Arsenal manager, according to Betfair, with Brendan Rodgers also said to be in the running.
Rodgers has done a brilliant job with Leicester and there may be reticence to leave the project at the East Midlands club.
Rafa Benitez, Invincibles legend Patrick Vieira and Atletico Madrid head coach Diego Simone are all 10/1 shots, according to the bookmaker.
Next Arsenal manager odds
Massimiliano Allegri - 13/5
Brendan Rodgers - 3/1
Rafa Benitez - 10/1
Patrick Vieira - 10/1
Diego Simone - 10/1
Eddie Howe - 11/1
Ralph Hassenhuttl - 11/1
Freddie Ljungberg - 16/1
Sol Campbell - 17/1
Steven Gerrard - 17/1
Arsene Wenger - 40/1
Sam Allardyce - 50/1
*All odds from Betfair
The writing may be on the wall for Arteta with many fans and former players calling for him to go, even though he won the FA Cup last season.
It has been a dismal season where Arsenal face a battle to even finish in the top ten.
Speaking on talkSPORT Breakfast, Trevor Sinclair said: "As much as you want him to do well as a young manager, he's brought a trophy to the football club already, which to me is one of the biggest reasons why he's still in a job.
"You just look at the performances and the lack of consistency and probably the lack of discipline since he's been in charge.
"It's just been a bit underwhelming. The Arsenal fans wanted to welcome him back to the club as a success.
"I think he had a fantastic apprenticeship with Pep Guardiola, but this job is far too big."
The Europa League exit may be the breaking point with angry Arsenal fans gathering outside the stadium following the game.
talkSPORT reporter Alex Crook tweeted that the Emirates was placed on lockdown.
He wrote: "We have been locked in the car park compound as a handful of angry #AFC fans have stormed the entrance.
"Saw fans screaming through the railings at the team coach as I left as well."