The 18-year-old scored in extra-time to help the Hammers beat Shrewsbury Town 1-0 and advance to the fourth-round of the FA Cup.
After being recalled from his loan spell with Bolton last month, their was speculation he could return to Macron Stadium, but Burke prefers to stay in London.
“There are still some talks about me going back to Bolton, but I've put a spanner in the works now," said Burke after Tuesday night's heroics.
“I'll see what the gaffer wants to do. If I'm here, I'll give my best and, if not, I'll go on loan do that elsewhere.
“I would like to stay here and get game time, but that's not my decision."