Sarri's Chelsea future has been brought into question ahead of Sunday's Carabao Cup final against City following three defeats in six matches in all competitions.
"We're learning every day," said Barkley. "We are all top-class players and we all want to learn more. If you compare Pep Guardiola's first season [2016-17], they went through a bit of an up-and-down spell. When I was at Everton, we beat them 4-0.
"We have got ambition to be one of the best footballing sides in the world. We're one of the biggest clubs in the world.
"The style of play the manager wants is not where it is at now. Over time it will come. Bad days don't always stay. Good times are ahead.
"We are doing it every day in training. I am sure, with time, you will see how good it will be."
One more sleep!— Chelsea FC (@ChelseaFC) February 23, 2019
Here's a reminder of how we sealed our place at Wembley... pic.twitter.com/f8r3U90XBg
Barkley added: "Every manager has their own way of approaching games. We did similar things at Everton under Roberto Martinez. It's just a way of being organised in a game. It's only going to help us. Personally, I don't mind it.
"He tries to improve every player. Throughout his career, you can see at Napoli how well he did with the players he had, how much he improved them. We went through a bit of an up-and-down spell in the last couple of weeks, but we're confident we'll come through it and achieve success."
‘I don’t want to press them, I want to keep their minds very free.' - Maurizio Sarri— Chelsea FC (@ChelseaFC) February 23, 2019
"Over the years – me not being at the club – you see managers come, managers go," Barkley said. "It's all about success, winning trophies. That's why I came to the club. If things aren't going well there's always going to be pressure that comes with it."