The former England striker rejoined the Hammers earlier this month on a short-term deal, having been released during the summer after seven years with the club.
Cole came on as a substitute in Sunday's goalless draw at Swansea and with Ricardo Vaz Te joining Andy Carroll on the treatment table, the 29-year-old looks set to lead West Ham's attack at Turf Moor.
"The next game we've got is Burnley and hopefully we can get goals there," said Cole.
"Everyone knows I need minutes for match fitness. Sunday was a good start on my way to recovery. I'm now hoping to get the minutes and the goals to get us through to the next round if I can.
"It was really good to be out there again. I was disappointed we didn't win but we were away from home so you can't ask for much more than getting a clean sheet.
"The boys have been doing well and we've got to give credit to them for only conceding one goal away from home so far this season. It's a huge credit to all the playing staff and the management as well.
"I'm chuffed to get back and to be involved in it again. Any footballer will tell you that have to work hard and the manager has noticed that I was working hard and has given me a new deal.
"Hopefully I can push on from here and add to the team and play a significant role."