Moyes was relieved of his managerial duties on Tuesday morning following a disastrous first season in charge of the Premier League champions.
"I think everyone's very disappointed with the way it's happened - all in the press first. Tempers and feelings are running high but Manchester United are trying to do the best by Manchester United," he told Sky Sports News.
"Throughout the season, it's been documented that there have been a few problems with a few of the players. It looks like they've never been addressed. You can't really have ill feelings towards your players.
"When you go into a football club like United, of course you're going to get the backing of your players because they want to be winning things season in, season out. This season, it hasn't gone for them and everybody's been disappointed with that."
Cole scored 121 times in 275 games for United between 1995 and 2001.