The England international has spent over a decade with the Gunners and is the longest-serving and top scoring player in the current Arsenal squad.
However, the 28-year-old's time at the Emirates Stadium looks to be coming to an end, with the Toffees already in discussion over signing the former Southampton forward.
"It's not imminent but negotiations are going on and we're hoping to negotiate a permanent deal," said Allardyce.
"I don't get optimistic otherwise I get disappointed. I'll be very excited and enthused when someone, even if it's not Theo, has signed on the dotted line."