Moyes said in December that Welbeck needed to follow the example of Wayne Rooney and stay behind after training to improve his finishing.
However, the England international insists that he has always chosen to do additional drills and suggested that Moyes must have missed him staying behind.
"I was pretty surprised [at that] to be honest. Before he made that statement I was doing extra training. Maybe he just didn't see it," Welbeck is quoted as saying by PA.
"I've grown up at Manchester United and been professional all through my career and I always do extra work. It's born and bred in me to do extra work after training and maybe he wasn't seeing that extra stuff. I was doing extras in the gym and on the pitch."
Moyes was sacked as United manager in April following a tumultuous 10-month spell at Old Trafford.