The loss means United have been beaten in three of their first four Premier League home games this season and slump to 15th place in the table, the side's worst start to a league season in seven years.
Speaking on Sky Sports, United legend Roy Keane was downbeat about Solskjaer's prospects.
— Sky Sports (@SkySports) November 1, 2020
"At this moment in time it's not looking good for him,” said Keane.
"Ole will lose his job working with these players, as night follows day - that's what is going to happen", Keane added after seeing the Gunners win at Old Trafford for the first time in 14 years.
But the Red Devils have been poor in the Premier League this season.
They lost 3-1 to Crystal Palace on the opening day. They then were hammered 6-1 by Tottenham before drawing at home against Chelsea.
Against Arsenal, it was a similar story.
United's next two games are against Champions League new boys Istanbul Basaksehir tomorrow afternoon followed by Carlo Ancelotti's Everton on Saturday lunchtime, it what is shaping up to be a crucial test for the United manager.
Should Solskjaer leave Old Trafford then former Tottenham boss Mauricio Pochettino is odds-on favourite to replace him at 8/13.
Other likely contenders include former AC Milan and Juventus coach Massimiliano Allegri at 6/1, Southampton boss Ralph Hasenhuttl at 12/1 and former PSG manager Laurent Blanc at 14/1.
Some United fans are already seemingly looking forward to Pochettino's predicted arrival at Old Trafford.
— Jonny Ross (@thejonnyross) October 29, 2020