Chelsea will begin their Champions League campaign in October and have automatically qualified for the group stages after winning the WSL last season.
Bompastor said her successor Hayes has left Chelsea in a “great place” and believes the club have all the tools to go one step further and win a European title.
“They were so close last season to being in the final of the Champions League,” she added. “I will try to bring my experience and winning mentality to make the next step, but there is no pressure on that.
“If you want to be honest, when you come to this club you get everything you need to perform. The club has all the infrastructures we need to perform, they have the players, they have the quality in the staff.”
Bompastor was under contract with Lyon until 2025 and said it was a big decision to leave the club where she had so much success as both a player and a manager.
“I really love to be challenged,” she said. “It’s been part of my life when I was a player and even when I’m a coach. I need to be challenged every day, every week.
“This is why I love doing this job. Coming to Chelsea is a big move, but this is a place where I wanted to be because I could feel the ambition from the sporting directors and all the club. I just felt this is the place where I want to be.”