The 28-year-old, at the Emirates Stadium for more than a decade, has been tipped to depart this month after falling down the pecking order.
Everton appear to be at the front of the queue for the Englishman's signature, though, as Allardyce has already made an official approach.
"If we're able to get that over the line, I'll be delighted," he told reporters. "If we can get that all sorted out then that would be a great addition to join us in an attacking point of view.
"There are two or three reasons [for interest]: goalscoring threat, pace, experience and he's good on assists from wide in terms of his crossing.
"So if that's at all possible, that would be very nice. It's not imminent, but negotiations are going on."
Walcott has been used a further nine times in other competitions, playing a direct part in six goals.