The striker, 30, rejoined the club in the summer on a short-term deal after originally being released on a free transfer.
Cole scored in the win against Fulham on Saturday and now wants to prove that he deserves to stay at Upton Park.
"Once you score you feel you can get in a rhythm and score more so hopefully there will be more to come for me now," Sky Sports News quotes Cole as saying.
"Hopefully they will be important goals. I want to keep the momentum and doing the right thing.
"I want to stay, of course, and I'm very confident. I know the ins and outs of the club. Sometimes change is good but I'm here and have been here for so long - I'd like to stay on."
West Ham are currently 15th in the Premier League.