The Hammers have endured a testing 2013/14 campaign, with inconsistencies leaving them sweating on their Premier League status at various stages.
A 2-1 victory over Sunderland on Monday would appear to have ended any lingering doubts regarding their top-flight standing, with an 11-point gap opened up on the bottom three.
The Upton Park outfit now have their sights set on a top-half finish, with the club determined to look upwards rather than over their shoulder.
Nolan is convinced that Allardyce, who was disappointed to see his side jeered by their own supporters following a recent 2-1 victory over Hull City, can guide the club into a brighter future, with there no reason why they cannot be pushing the top six in the years to come.
He said in the Daily Star: “Of course he’s had a difficult week.
“With all the online stuff that goes on, it can go really stupid. But Sam’s got a good rapport with the fans and I think they know that he knows what he’s doing.
“If he’s given the armoury he wants, he can build a team that will be challenging for Europe – and doing it the right way.
“You only have to look back at what we had at Bolton, when he was given the ammunition to go on and the situation he got us in.
“We qualified for Europe twice and when he left, they struggled.”