Chiesa, who scored in Liverpool's 4-2 pre-season win over Sunderland in Nashville on Saturday, made 36 appearances in all competitions last season but started only five matches as he struggled to establish himself in Slot's plans.
The 28-year-old was linked with a return to Serie A after last season, but said his priority was to work with Iraola, who succeeded Slot in June, and earn more opportunities at Anfield.
"It is always a fresh start at the beginning of the year. Maybe there have been too many fresh starts. But I don't care, I am trying to do my best for this new manager," Chiesa told reporters.
"Last season under Arne Slot, I was trying to do my best. Last year I felt ready for a bigger role, then it depended on the coach whether he wanted me to play or not.
"Of course I wanted to play more but it is what it is. The gaffer had his gameplans, he took his decisions, I have nothing to say about that. It is football. This year, let us say it is a new chapter. There is a new manager and I have to focus on that."
Chiesa, who joined Liverpool in 2024 and is under contract until 2028, indicated in an interview with La Gazzetta dello Sport in June that he could consider his options if he was unable to secure regular playing time.
However, the Italy international said he remained committed to Liverpool and was concentrating on his performances under Iraola.
"Right now, at the moment, I am happy here at Liverpool. I love the club, the fans, everything," he said.
"I am doing the best I can to have a chance here - and then we will see. At the minute, the only thing I am thinking about is Liverpool."