Moyes was sacked as United boss in April, just 10 months into a six-year contract.
The Scot claimed last December that Welbeck was failing to follow the example of Wayne Rooney, who regularly stayed behind after training to improve his shooting.
"We had a word with (Welbeck) about a month ago and we said that he needs to be the last off the training field," Moyes told a press conference on December 17.
"Wayne's out there practising his finishing each day... (and) I said: 'Danny, you need to be out there every day finishing, even if it's 15 minutes at the end'."
Moyes may have intended his words as constructive criticism, but Welbeck did not take them well.
"I was pretty surprised (at that) to be honest," Welbeck said. "Before he made that statement I was doing extra training. Maybe he just didn't see it.
"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."
Welbeck, who scored 11 goals in 24 starts last season, was clearly unhappy at United during the campaign.
Moyes asked the England man to play out on the left on a number of occasions, although it is also worth pointing out that his predecessor Sir Alex Ferguson fielded Welbeck there too.
Welbeck said: "It does get frustrating. You want to be playing in a certain position and you're not getting the opportunity to do that.
"It's the same for everyone - when they are being played out of position they don't really like it.
"I'd like to play central. I've been playing on the left for a while and it's got to the time where I want to stake a place up front.
"My goals per game ratio is better for England. I'm much freer up front as there are three in midfield and three up front. That way there's a bit more defensive cover rather than me being that defensive cover.
"For United I'm more likely to be left wing of a four-man midfield with only two centre-mids so it's a bit more difficult for me to maybe make those runs to get in and score those goals when I've got to think about my defensive duties as well.
"Maybe I'm kind of let off the leash a bit more for England than United."
It emerged that Welbeck was considering leaving the club he has supported all his life towards the end of last season.
Competition for the striker may increase if United sign another front man this summer.
When asked whether he wanted to leave Old Trafford, the United academy product said: "I'd rather not answer that question."