There was a lengthy lull in play in the second half of the Premier League clash as substitute Potts received treatment on the pitch after a collision left him unconscious on the ground, with 12 minutes' injury time subsequently added.
Allardyce says the 18-year-old will be kept under observation after being stretchered from the pitch and taken to hospital.
He told Sky Sports: "He's not good. He's been taken to hospital to keep him under observation.
"Hopefully it's just a concussion and nothing more than that. Hopefully it's no more severe than that.
"Obviously he didn't really know where he was. He had a bad bang on the head to start with and landed with his head hitting the ground."
Turning his attention to the disappointing result at Emirates Stadium, Allardyce admitted West Ham were undone by a scintillating 10 minutes from Arsenal in the second half during which they scored four goals.
The manager said: "Unfortunately for us there was 10 minutes of brilliance from Arsenal that we couldn't compete with and that 10 minutes of brilliance brought four very good goals and that was the game done and dusted.
"After that for us it was really about making sure we didn't concede any more than that."
Allardyce added: "Those 10 minutes they blew us out of the water and you have to hold your hands up - they were brilliant.
"It's one of those things when you come here. You hope you can play your very best but when Arsenal are on their best form it's very difficult to get a result."