Blues are through to the last four of the competition for the first time in a decade and now face West Ham over two legs.
But they are hovering just above the Premier League relegation zone despite an upturn in recent fortunes.
Jerome told Press Association Sport: "The league is our bread and butter and the cup is a bonus. It's nice to get to the semi-finals and obviously we want to get to the final and Wembley.
"We've got a chance of winning it and anything is possible in football. But we've got to keep our feet on the ground, concentrate on the league and not slip into the relegation zone.
"A couple of defeats, and a couple of wins for other teams, and you can be right back down there."