The 31-year-old, who began his career at Upton Park in 1994, rejoined his boyhood club in January and then helped fire the Londoners to a top-10 finish in the Premier League last season.
Cole now wants West Ham to finish even higher in the league in 2013-14, but his main ambition is to take the club to Wembley.
"I think the ambition of the club at the moment is improvement," he told reporters.
"But for me personally, I would love to finish better than 10th but I would 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.
"You look at Swansea and Wigan last year and that is their day, we would like a bit of that because it has been such a long time since West Ham have been there, so that would be nice."
Cole was on the scoresheet on Sunday, as the Hammers kicked off their pre-season preparations with a 6-2 win over Cork City.