The Red Devils have moved two points ahead of the Gunners and are now just three points off a Champions League qualifying spot - with a game in hand.
Arsenal have qualified for Europe's elite competition every year since Wenger took charge in 1996 - but they are now in danger of falling short of a Europa League spot.
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Arsene Wenger lost 3-1 to Tony Pulis' West Brom
Wenger, 67, is out of contract at the end of the season and admitted after the West Brom defeat that he has made a decision on his future
"I know what I will do," said Wenger. "You will soon know.
"Today I do not necessarily worry about [my future]. We are in a unique bad patch we never had in 20 years.
"We lose game after game at the moment and that for me is much more important than my future."
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