The England striker is yet to feature this term due to an ankle injury he suffered in the summer, but is now back in training.
In his absence, the Hammers have beaten Liverpool and Manchester City - Premier League runners-up and winners in 2013-14 - and are fourth in the standings after nine games.
"I think we can push right on for Europe," Carroll told whufc.com.
"I don't see why not - I think all the lads are positive about that and I know the gaffer is too.
"We're playing fantastically
The gaffer has signed some great players and we're showing the quality we've got with the goals we're scoring
That's why we are where we are in the league."
Allardyce has often been criticised for a direct approach, with targetman Carroll key, but the likes of Enner Valencia and Diafra Sakho have given the Hammers an added dimension.
Carroll has also been keenly watching Aaron Cresswell's performances at left-back and is relishing the chance to work with a player who has delivered accurate crosses time and again this term.
The 25-year-old former Newcastle and Liverpool striker added: "We scored three headers at Burnley, and another on Saturday (against City) with Cresswell crossing the ball in
He whips them in and puts them on a plate.
"The service that everyone is putting in the box to create chances has been great
That's what I thrive on
I can't wait to be involved with that."
Source : PA
Source: PA