Cole started in the opening two games of the season against Aston Villa and Swansea before being replaced by Carroll after he completed a late loan move from Liverpool.
The 57-year-old is hoping the striker can find the form that helped the club reach the Premier League last season.
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|7/1||Carlton Cole is 7/1 with Paddy Power to score the first goal against Norwich on Saturday
Allardyce told reporters: "He is an obvious replacement and we would like Carlton to do how he finished off last season.
"He scored a hatful of goals for us and helped us get back into this lucrative Premier League so we would like him to step in and start scoring the goals he did towards the back-end of last season.
"We hope he can hit the goalscoring trail when he gets the opportunity on Saturday."
The Hammers boss also remained unclear on when Carroll would be able to make a comeback, but was confident the striker wouldn't take any longer than six weeks to recover.
Carroll has been aiming to play in the London derby against Arsenal on October 6, meaning a five week absence.
Allardyce added: "We have got no time on it really. It will be a maximum of six weeks but he is improving well so we continue day-by-day and look at every day and see what type of improvement we have got and go from there.
"If there is a possibility we can get him back quicker we will be trying our best."