The Hammers are currently second-bottom of the Barclays Premier League, have won only one of their last 13 top-flight fixtures, and on Sunday suffered a humiliating 5-0 defeat to Sky Bet Championship outfit Nottingham Forest in the third round of the FA Cup.
In an open letter published on West Ham's official website on Monday, Sullivan and Gold signalled their support for Allardyce and confidence that the club's fortunes will turn around.
And Cottee told Press Association Sport: "They have done the right thing in backing him, because if you are not going to do that, you might as well get rid of the manager.
"With any manager you have to support them.
"But Sam doesn't need me to tell him that he needs results and needs them very quickly."
He added: "There is no doubt the club is in trouble. I don't think it's a full-blown crisis yet, but I think we're not far off it.
"The bottom line is that you need to get results in football, and the results in the Premier League have not been good recently, plus they have just got knocked out of the FA Cup.
"The rest of January - the next four or five games - are vital for the club, and if they don't get the results, it is going to be a full-blown crisis."
West Ham have a host of players unavailable for selection at the moment, including the suspended Kevin Nolan, injured Winston Reid and Andy Carroll, their most notable signing of the summer, who has spent the entire campaign so far sidelined due to foot problems.
The Hammers are at Manchester City on Wednesday in the first leg of their Capital One Cup semi-final, and with a view to that match and forthcoming fixtures in the Premier League, Allardyce opted to field a vastly inexperienced team at Forest.
He suggested afterwards that he had little choice and the co-chairmen's statement has indicated their understanding of the decision.
Cottee can see why Allardyce made the selection he did, but regards the manner of the Forest defeat - in which some West Ham regular first-team players did feature - as "unacceptable", and thinks Hammers fans are owed a decent performance in the City game.
"I understand what Sam is saying and the reasons for bringing in so many kids," said Cottee, who had two separate spells playing for the club, one in the 1980s and the other in the 1990s.
"But I think the manner of the defeat was unacceptable - almost like a passionless performance.
"I felt for the fans. I think they were just shell-shocked. I was, watching on the television. I don't think anyone could believe it.
"I know not everyone who was playing yesterday will be playing on Wednesday, but there will be some, and they need to put things right with the fans."
Asked if he thought Allardyce was the right man to get West Ham out of trouble and should remain as their boss, the 48-year-old ex-England frontman added: "I can't answer that, because there have been so many injuries and suspensions, which have meant he can't put his strongest team out.
"It is hard to judge Sam as a manager at the moment, because his best striker (Carroll) has been out all season and his best centre-half, Winston Reid, is missing now.
"It is hard to judge whether we are going in the right or wrong direction overall.
"But in the short term, it is important that everyone gets behind the manager and that above all else, the players put in performances.
"Sam can do a Winston Churchill speech before a game, but if the players don't go out and perform, it is not going to make any difference.
"I think there is enough talent in the squad for them to stay up.
"If everyone is fit, I think we have as good a squad as any of the other teams that are down at the bottom. So yes, they can get out of it.
"As for whether the players are behind the manager or not - I think we'll find out in the next two or three games, because it is alright having a good team spirit and saying the right things, but you have to do your talking on the pitch."