Cole finally netted his first goal of the season in last Saturday's 3-1 comeback victory over Chelsea, while fellow striker Andy Carroll has also scored just once for the Hammers.
Allardyce admits his misfiring forward-line is one of his few concerns about a side which is sitting just outside the top six, but he has challenged Cole to produce a goal-spree ahead of the trip to Liverpool.
He said: "It was a great comeback in terms of the fact he has waited patiently and I think his performance overall, including the goal and the time he scored it, was obviously a key element in us going on to win the game.
"I hope he can carry it on and continue to score goals because that has been one of our slight weaknesses - that our centre forwards haven't really scored enough.
"Everyone else has because we're one of the best clubs for sharing out goals in the Premier League.
"We've had lots of players scoring goals but we've not had anybody rattling the goals in apart from Kevin Nolan.
"Our great win came from clinical finishing and we want to continue that if we can."
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