André Villas-Boas’ men have improved since a shaky start to their Premier League campaign and the north London outfit have won seven of their last nine top-flight games.
Tottenham’s terrific run has seen them leapfrog London rivals Chelsea and move into third position, just six points behind the defending Premier League champions, who are further seven points behind leaders Manchester United.
“That’s the goal [to finish second],” Dempsey told PA.
“Everybody wants to be pushing for great things and doing something special in their careers. We have the quality in our team and there’s no reason why we shouldn’t think that.
“We need to continue on this good run of form and get as many points as possible.
“You want to keep your feet on the ground and know there’s a lot of hard work to put in if you want to accomplish your goals.”
Dempsey has scored six goals for Tottenham after completing a £6m move from Fulham to White Hart Lane on the final day of the summer transfer window.
The American endured a muted start to his Spurs career, but Dempsey’s form has improved in recent weeks, with a return of two goals and three assists in his last eight outings.
“My season is going OK,” Dempsey added.
“I’m someone who likes to score or get assists so it’s good to be getting those stats up, contributing and trying to help this team do well. It’s going well but I still want to be better at what I’m doing.”
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news