BBC broadcaster John Murray noted the veteran coach will "soon" disclose whether he will stay in north London or depart his role.
It led to Wenger being asked questions again regarding his future - with his contract set to expire at the end of the season.
More poor marking, with Arsenal players positioned like statues for a James McLean corner, allowed the towering Dawson to seal West Brom's win with a thumping close-range header in the 75th minute.
Wenger revealed Sanchez had been substituted with an ankle injury, which will be assessed before the club decide whether to withdraw him from the Chile squad.
Arsenal need fresh enthusiasm and ideas, and a change of manager will aid their recovery next term.
It was an eventful encounter, both on the pitch and above it as two planes pulling banners - one pro-Arsene Wenger and the other against the Arsenal boss - made circles around The Hawthorns.
Wenger's future beyond the end of this season remains unknown, but the facts are these - Arsenal trail fourth place by five points (albeit with a game in hand).
"I know what I will do in my future, so you will soon know, very soon", he said. "It's more. It's not that", he said.
"We are in a unique bad patch which we've never had in 20 years you know", Wenger said. "Financially, I mean. Of course, on the sporting front it would be a blow, but financially the Champions League does not have the impact any more that it had five or six years ago because of the influx of the television money (to the Premier League)". "That's a great deal for Arsenal Football Club", said Pulis.
"I take a bigger perspective than that".
"In the end we had to take him off". I'm judged enough and I'm not a judge so I can't answer.
The Sunday World says Wenger is expected to tell the Arsenal hierarchy that he wants to remain as the club's manager next season, with talks planned in the next few days.
"He stayed on, but in the second half he was not himself so I made a decision to take him off".
After the global break, Arsenal's next Premier League game is against Manchester City, the side they face in next month's FA Cup semi-final.
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