West Ham were winning 1-0, they were at home and Hull City were down to 10 men. Everything was going to plan on the surface but on further investigation, fault lines were apparent around Upton Park, with the mood of frustration captured by an interview at half-time with George Parris, a member of the side that finished third in 1986. "Well, I'm enjoying the score," Parris said when he was asked if he was enjoying the match and it was not hard to read between the lines. In the stands, there were nods and wry smiles. West Ham's fans knew what he was getting at: they were bored too and the boos at the final whistle reflected that.

Sam Allardyce, cupping an ear towards the crowd in astonishment, could not believe what he was hearing given that West Ham had secured a 2-1 victory that virtually ensured their Premier League survival. It was a strange time to be voicing disapproval and it is a sorry state of affairs when winning is no longer good enough.

"For all my criticism of Sam Allardyce, when you start making your home ground that hostile when you win – and I say hostile in inverted commas because it maybe wasn't as hostile as we've known it in the past – you don't look forward to coming back home," the former West Ham goalkeeper Shaka Hislop says.

While that is true, the booing was indicative of Allardyce's unpopularity with the West Ham support, many of whom perceive him to be overly negative. "It is extraordinary when it gets boring," Phil Whelans, the presenter of the STOP! Hammer Time podcast, says. "He drills a defence really well but his attacking ambition is sort of zero."

West Ham are unsure whether to stick or twist. The misgivings over the style of football mean that the board is considering whether to sack Allardyce in the summer and hire a more attacking manager. Yet Allardyce has never been relegated and the move to the Olympic Stadium in 2016 makes another drop into the Championship unthinkable. The stakes are so much higher than when Avram Grant took West Ham down in 2011.

There was more of a case to sack Allardyce when West Ham, ravaged by injuries and poor planning in the transfer market, were struggling earlier in the season. Yet they stuck by him and six wins in their past nine matches have moved them away from danger. Monday's gritty 2-1 win at Sunderland edged them closer to safety.

David Gold, the co-owner, has backed Allardyce but behind the scenes there is acknowledgement of the criticism. No fair assessment of Allardyce can conclude that he has done anything other than a fine job at West Ham, winning promotion from the Championship at the first attempt after replacing Grant in 2011 and then finishing 10th last season, while he has made the team resilient. West Ham have kept 13 clean sheets this season and Allardyce's tactics served them well in January's 0-0 draw at Chelsea. Only José Mourinho complained about "19th-century" tactics that night.

Yet West Ham's fans want more from the Academy of Football. Allardyce had similar problems at Newcastle, while Alan Curbishley was also considered to be too dull during his brief spell at West Ham. "This is something that Sam is always going to be battling," the former West Ham striker Tony Cottee says. "The seeds for that were sown a long time ago. When you appoint Sam Allardyce, you know what you're going to get. I think he deserves credit in terms of getting the club promoted at the first attempt, following on from what was an awful period under Avram Grant. The club was in chaos when Sam arrived, so it was good they got back straight away and finished 10th last season, which was fantastic. Sam's done what he says he does, he doesn't get teams relegated."

Yet it is a fragile alliance. "I maintain that I didn't think Sam was the right choice," Hislop says. "Give me the Ron Greenwood traditions of West Ham. But at the same time, promotion and then survival was more important financially to the long-term survival of the club. You could always regain those traditions. So I was kind of happy to see Sam as manager but always felt it was not going to be a long-term solution."

From Allardyce's point of view, West Ham beat Hull and nothing else matters. He has delivered results and as much as Allardyce is told about the "West Ham way" – occasional inspiration mixed with long spells of incompetence – he has succeeded using his methods. When he was criticised for a lack of entertainment in 2012, his response was that no one could tell him what the "West Ham way" was.

Allardyce, whose comments can be antagonistic, is resigned to the perception of him as a long-ball merchant but believes that his players have not received enough credit. "It's all a load of bull, isn't it?" he says. "I can't help bull. I think that it's all about the perception and the reputation of Sam Allardyce, not the West Ham players and how they play. It's all perceived to be like this and nothing else. It is a load of rubbish and I can't help that. We've got limitations, we know that, but we've also got some very good and very creative players."

West Ham's squad is not packed with cloggers and it is not beyond them to play attractive football, yet their attacking approach is often centred around Andy Carroll's strength in the air and Kevin Nolan's instinct for scrappy goals. West Ham tend to create pressure rather than chances and the failure of the Hull game was their paucity of ideas against a side with 10 men.

"I can see both sides of the argument," Cottee says. "I can certainly see where Sam's coming from. I'm not sure his actions at the end of the Hull game were the right thing to do but I understand why he did them and he's right, to be booed having won the game is rather unusual. At times there has been some entertaining football.

"The older fans have been brought up on the legendary Moore, Hurst and Peters team and then you had the teams that won the FA Cup in '75 and '80 and then the '86 team that I played in. The old fans were brought up on the team playing good football. They might not necessarily have ever really been close to achieving it consistently but they always strived to be playing good football and that came from Ron Greenwood and then John Lyall."

The move to the Olympic Stadium increases the pressure to stay up and although it is difficult for supporters to take a long-term view, Allardyce comes with a safety guarantee.

"West Ham can't really afford to get relegated," Cottee says. "It would be very dangerous to remove Sam because the club needs stability. I want West Ham to be competing with Tottenham, Chelsea and Arsenal. I want to see West Ham go on to the next level and I think they're in danger of being left behind if they don't move."

Whether Allardyce is the man to lead them into the Olympic Stadium remains debatable, but he may be the man to guide them towards it. Perhaps West Ham's fans should be careful what they wish for.