West Ham have finished in 13th and 12th place respectively in the last two years, but Slaven Bilic has transformed the Upton Park outfit into a side who have ambitions of qualifying for Europe ahead of their switch to the Olympic Stadium.
The East Londoners currently sit in seventh position in the table - ahead of the likes of Liverpool, Everton and Chelsea - while they are just a point adrift of Manchester United, who are in fifth position.
Winterburn, who made 94 appearances for West Ham between 2000 and 2003, has insisted that remaining in seventh place should be considered a positive achievement, but he has backed Bilic's men to aim higher.
When asked what would be considered an acceptable position, the 52-year-old said: "Anywhere above where they are now. They’ve done well away from home, got a fantastic squad.
"They will definitely finish high up the league. I would say anywhere between seventh to fifth would be a good season for them."
As well as losing just six times in 26 league games, West Ham are enjoying a run in the FA Cup, with last weekend's 5-1 demolition of Blackburn Rovers earning the club a place in the quarter-finals of the competition.
Before Bilic's arrival, West Ham had only reached that stage of the tournament once in the past nine years.
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