The Hammers last won a major trophy in 1981 when they lifted the League Cup, but also lost in the final of the FA Cup in 2006 to Liverpool.
"I think the ambition of the club at the moment is improvement," Cole told the club's official website.
"But for me personally, I would love to finish better than 10th, but equally I'd love to take this club to Wembley. Because at the end of the day, they are the days you remember in your career - it would be a massive achievement.
"West Ham are a big club but have been starved of success. You look at where we are in London and people want to come here and play in London and play for West Ham."
Cole rejoined West Ham last season for his second spell at Upton Park, 10 years after leaving the club to join Chelsea.